geometry problems from Kyiv TSTs for the Ukraine Mathematical Olympiad with aops links in the names
collected inside aops: here
2005-21)
most problems are not original
Let $AD$ be altitude, $BE$ be median, $CF$ be angle bisector of acute triangle $ABC$. Denote $M,N$ the points of intersection of $CF$ with $AD$ and $DE$ respectively. Find the perimeter of the triangle $ABC$ if $FM=2, MN=1$ and $NC=3$.
Let $I$ be the center of the circle inscribed in a triangle $ABC$. Let $w_1$ be a circle passing through point $B$ and touching $CI$ at point $I$. Let $w_2$ be a circle passing through point $C$ and touching $BI$ at point $I$. Prove that one of the points of intersection of the circles $w_1,w_2$ lies on the circle circumscribed around triangle $ABC$.
The desert is divided into square cells, infinite on all sides one of which is a lion. The hunter sets up fences, each of which divides $2$ neighboring cells of the desert, and the lion after setting each subsequent fence runs to one of the neighboring cells (ie with a common side, while the lion can not jump over the fence). In order to hunt a lion, the hunter must completely enclose some cell of the desert, which is a lion. Can he do this with restriction to install $100$ fences?
(Bryman W.)
Let $ABCD$ a parallelogram. Let $E,F$ be the points of intersection of the altitudes of triangles $ABC, BCD$ respectively. Prove that $\angle CAF=\angle BDE$.
(Bryman W.)
Let $ABC$ be a triangle with $\angle C=60^o$ and $AC<BC$. On the side $BC$ mark the point $D$ in such a way that $BD=AC$, and on the extension of the $AC$ beyond point $C$ mark the point $E$ so that $AC=CE$. Prove that $AB=DE$.
Two circles with centers at points $O_1,O_2$ intersect at points $A,B$. A circle is drawn through the points $O_1,O_2,A$ which intersects the given circles at the points $K,M$ for the second time. Prove that $AB$ is the bisector of the angle $\angle KAM$.
(Timoshkevich T.)
Let $ABCD$ be a square, point $E$ lies on the side $CD$, $O$ be the center of the square, $N$ be the midpoint of $BC$, $P=BE \cap NO$. It is known that $\vartriangle POE$ is isosceles. In what ratio does the point $E$ divide the side of the square $CD$?.
(Klurman O.)
Let $I$ be the incenter of $\vartriangle ABC$. It turns out that $CA+AI=CB$. Prove that $\angle CAB=2\angle ABC$
Given a triangle $ABC$ with $AC=BC$. Take a point $P$ on the one of the two arcs $AB$ of the circle circumscribed around this triangle, on which the vertex $C$ does not lie. Let $D$ be the foot of the perpendicular drawn from the point $C$ on $PB$. Prove that $PA+PB=2PD$.
Chords $UV$ and $RS$ of the circle with the center $O$ intersect at a point $N$. Let $AU$, $AV$, $BR$, $BS$ be tangents to this circle. Prove that the lines $ON$ and $AB$ are perpendicular.
Let the inscribed circle in the circle $ABC$ touch the sides $BC,CA,AB$ at the points $D,E,F$ respectively. Take a point $K$ inside this triangle such that the inscribed circle in the triangle $KBC$ also touches $BC$ at the point $D$. Let this circle touch both $CK$ and $KB$ at points $L$ and $M$ respectively. Prove that the quadrilateral $EFML$ is cyclic.
The points $A,B,C,D,E$ were marked on the circle (they are located on the circle in this order). Rays $AE$ and $CD$ intersect at a point $L$. The point $K$ chosen on the ray $AC$ such that $BK$ is tangent to this circle. It turned out that the line $KL$ is parallel to the line $ED$. Prove that $KB=KL$.
The points $A,B,C,D$ were marked on the circle (they are located on the circle in this order). Tangent to circle at point $A$ intersects ray $CB$ at point $D$, and tangent to circle at point $B$ intersects ray $DA$ at point $E$. It is known that $AD=AE$ and $BC=BD$. Prove that $ABCD$ is a parallelogram or a trapezoid.
On the hypotenuse $AB$ of an right isosceles triangle $\vartriangle ABC$ we mark points $D$ and $E$ are such that $\angle DCE = 45^o$, $AD = 8$ and $BE = 9$. Find $AB$.
Let $ABCD$ be a trapezoid, circle $\omega_1$ with center $O_1$ inscribed in triangle $\vartriangle ABD$, and circle $\omega_2$ with center $O_2$ touches the side $CD$ and extensions of the sides $BC$ and $BD$ of the triangle $\vartriangle BCD$, and $AD \parallel O_1O_2 \parallel BC$. Prove that $AC = O_1O_2$.
(V. Yasinsky)
A circle can be circumscribed around the quadrilateral $ABCD$. Points $P$ and $Q$ were marked on the sides $AB$ and $AD$ respectively such that $AP = CD$ and $AQ = BC$. Let $N$ be the midpoint of $BD$. Prove that $PQ = 2CN$.
On the circle $\omega$ the points $A, B, C$ and $D$ are chosen so that the tangents to the circle $\omega$ on the points $A$ and $B$ and the line $CD$ intersect at point $K$. On the lines $AC$ and $AD$, the points $E$ and $F$ are chosen, respectively, so that the line $EF$ passes through point $B$ and $EF \parallel KA$. Prove that $BE = BF$.
Let $I$ be the center of a circle inscribed in triangle $\vartriangle ABC$. The lines $AI, BI$ and $CI$ intersect the circle $\omega$, circumscribed around the triangle $\vartriangle ABC$, the second time at points $D, E$ and $F$, respectively. Let $DK$ be the diameter of the circle $\omega$ and $N$ be the point of intersection of $KI$ with $EF$. Prove that $KN = IN$.
(T. Timoshkevich)
A straight line is given on the plane, on which $2n$ points are selected. Another $n$ points are selected outside this line. Is it always possible to construct $n$ triangles without common points with vertices at given points?
(B. Rublev)
Consider the triangle $ABC$ and the point $D$ belonging to the side $BC$. Denote by $P, Q$ the points of intersection of altitudes of triangles $ABD$ and $ADC$. For which points $D$ are the triangles $ABC$ and $DPQ$ similar?
(B. Rublev)
The pentagon $ABCDE$ is inscribed in a circle, $AC \parallel DE$ and $M$ is the midpoint of the diagonal $BD$. Prove that if $\angle AMB = \angle BMC$, then the line $BE$ divides the diagonal $AC$ in half.
(V. Yasinsky)
Let $O$ be the midpoint of the side $AB$ of triangle $\vartriangle ABL$. Perpendiculars bisectors drawn to the segments $AO$ and $BL$ intersect at point $V$, and the perpendiculars perpendiculars drawn to the segments $AL$ and $BO$, intersect at the point $E$. Prove that $LO\perp VE$.
On the extension of the side $BC$ of the triangle $\vartriangle ABC$ beyond the point $B$ draw the segment $DB = AB$. Let $M$ be the midpoint of the segment $AC$ and let $P$ be the point of intersection of $DM$ with the bisector of the angle $\angle ABC$. Prove that $\angle BAP = \angle BCA$.
Find the number of lines that intersect some two sides of a given right triangle and the circle inscribed in it , successively at the points $L, O, V, E$, and $LO = OV = VE$.
(W. Bryman)
A circle is inscribed in an acute angle with vertex at point $A$, tangent to the sides of the angle at points $B$, $C$. Prove that the length of an arbitrary segment, completely located inside the region, bounded by segments $AB$, $AC$ and the smaller circular arc $BC$, is no more than $AB$.
An altitude $CF$ is drawn in an acute triangle $ABC$ and median $BM$, and it turned out that $CF=BM$ and $\angle MBC=\angle FCA$. Prove that $AB=AC$.
Let $ a $, $ b $, $ c $ be the lengths of the sides of a triangle.
Prove the inequality:
$$ 2 <\frac {a + b} {c} + \frac {b + c} {a} + \frac {c + a} {b} - \frac {a ^ 3 + b ^ 3 + c ^ 3 } {abc} \le 3. $$
$ ABCD $ is a trapezoid with bases $ AB $ and $ CD $, its diagonals intersect at the point $ E $. The point $ X $ is the midpoint of the segment, connecting the orthocenters of the triangles $ BEC $ and $ AED $. Prove that $ X $ lies on the perpendicular drawn from the point $ E $ on the line $ AB $
$ABC$ is an acute triangle in which $\angle BAC>\angle BCA$. Let $D$ be a point on the side of $AC$ such that $|AB|=|BD|$. Next, let the point $F$ on the circumscribed circle of triangle $ABC$ such that line $FD$ is perpendicular on side $BC$, and the points $F$ and $B$ lie on opposite sides of line $AC$. Prove that line $FB$ is perpendicular to $AC$.
An isosceles triangle $ABC$ is given. Point $D$ is chosen on side $BC$ so that $BD=2DC$, and the point $P$ on the segment $AD$ so, that $\angle BPD=\angle BAC$. Prove that $\angle BAC=2\,\angle DPC$.
In the triangle $ ABC $ on the side $ BC $ the point $ X $ is taken. On the segment $ AX $. the point $ M $ is taken, and the lines $ BM $ and $ CM $ intersect with sides $ AB $, $ AC $, at the points $ T $ and $ P $ respectively. From the point $ X $ on $ TP $ the perpendicular is drawn. Find the locus of the feet of these perpendiculars.
Prove that there are three different lines, each of which passes through the midpoint of one of the sides of the triangle and divides this triangle into two polygons of the same perimeter, intersect at one point.
In a right triangle $ ABC $ with right angle $ \angle C $. the point $ M $ is the midpoint of $ BC $, $ I $ is the center of the inscribed circle. Let $ P $ be the intersection point of of $ IM $ and $ AC $, $ S $ and $ R $ be the touchpoints of the circle inscribed in $ \triangle ABC $ with the sides $ AB $ and $ AC $, respectively. $ Q $ is the intersection point of $ SR $ and $ BC $, $ L $ is midpoint of $ PQ $, point $ F $ is projection of point $ C $ on $ IM $. Prove that the points $ F $, $ L $, $ S $ lie on one line.
Let $ABCD$ be a convex quadrilateral. Points $E$ and $F$ are the midpoints of the sides $AD$ and $BC$, respectively. The segment $CE$ intersects $DF$ at point $O$. Prove that if the lines $AO$ and $BO$ divide side of $CD$ into three equal parts, then $ABCD$ is a parallelogram.
Points $K$, $M$ and $N$ are midpoints of sides $AB$, $CD$ and $AD$ of convex quadrilateral $ABCD$ respectively, $L$ is midpoint of segment $AN$. The lines $AM$, $BN$, $CL$ and $DK$ intersect at one point $O$. Prove that the broken line $BOM$ divides the quadrilateral into two figures of equal area.
The triangle $ ABC $ is acute-angled. $ M $ is midpoint of the side $ BC $, the point $ P $ is chosen on the side $ AM $ so that $ MB = MP $. The point $ H $ is the foot of the perpendicular drawn from the point $ P $ on line $ BC $. From the point $ H $, the perpendiculars on the lines $ PB $ and $ PC $ are drawn , intersecting the lines $ AB $ and $ AC $ at the points $ Q $ and $ R $ respectively. Prove that the line $ BC $ touches the circle, circumscribed around the triangle $ QHR $, at the point $ H $.
Let $ABC $ be an acute-angled triangle. Find the locus of the centers of the rectangles, all vertices of which lie on the sides $\Delta ABC$.
Let $ABC$ be an acute-angled triangle. Determine if there is a point that is the center of three different rectangles with vertices on the sides of the triangle.
Let $ABC$ be an acute-angled triangle, for some interior point of the triangle $P$ denote by ${{O} _ {a}}, {{O} _ {b}}, {{O} _ { c}}$ centers of the circumscribed circles of triangles $PBC, PCA, PAB$, respectively.
a) Find the locus of such points $P$ for which the equality holds: $\frac {{{O} _ {a}} {{O} _ {b}}} {AB} = \frac {{ {O} _ {b}} {{O} _ {c}}} {BC} = \frac {{{O} _ {c}} {{O} _ {a}}} {CA} $.
b) For each point $P$ belonging to locus of point from (a), prove that the lines $A {{O} _ {a}}, B {{O} _ {b}}, C {{ O} _ {c}} $ intersect at one point.
Given a parallelogram $ABCD$, which has $AB> BC$. Denote by $K$and $M$ the points of tangency to the diagonal $AC$ of the circles inscribed in the triangles $ACD$ and $ABC$, respectively, and by $L $ and $N$ are points of tangency to the diagonal $BD$ of circles inscribed in triangles $BCD$ and $ABD$, respectively. Prove that $KLMN$ is a rectangle.
A circle is circumscribed around the square $ABCD$. On the smaller of the arcs connecting the points $C$ and $D$, some point $M$ is selected. Let $AM$ intersect the segments $BD$ and $CD$ at the points $P$ and $R$, and let $BM$intersect the segments $AC$ and $CD$ at points $Q$and $S$, respectively. Prove that $RQ$ is perpendicular to $PS$.
An isosceles triangle $ABC$ ($AC = BC$) is given. Let $P$ be a point inside the triangle such that $\angle PAB = \angle PBC$. Point $M$ is the midpoint of the side $AB$. Prove that $\angle APM + \angle BPC = 180^ \circ $.
Denote by $D $ the midpoint of the side $AB$ of an acute $\Delta ABC$. On the sides $AC$ and $BC$, the points ${A} '$ and ${B}' $ are selected, respectively, such as triangles $AD {A} '$ and $BD{B} '$ are isosceles with a common vertex $D$. Prove that if the lines $CD $ and ${A} '{B}'$ are perpendicular, then $\Delta ABC$ is isosceles.
In $\Delta ABC$, in which $AB <BC$ on the side $AC$ is selected a point $D$ such that $AB = BD$. The circle inscribed in $\Delta ABC$ touches the sides $AB$ and $AC$ at the points $K$ and $L$, respectively. Let $J $ be the incenter $\Delta BCD$. Prove that the line $KL$ bisects the segment $AJ$.
Let ${{l} _ {a}}, {{l} _ {b}}, {{l} _ {c}}$ be the lengths of the angle bisectors of $\Delta ABC $ with sides $a, b, c$, $R $ is the radius of the circumscribed circle. Prove the inequality:$$\frac{b^2 + c^2}{l_ {a}} + \frac{c^2+a^2} {l_b} + \frac{a^2 + b^2}{l_c}>4R $$
The two circles ${{w} _ {1}} $ and ${{w} _ {2}}$ touch externally. Their common external tangent touches the circles ${{w} _ {1}} $ and ${{w} _ {2}}$ at the points $A $ and $B$, respectively. Let $AP $ be the diameter of the circle ${{w} _ {1}} $, and let the tangent to the circle ${{w} _ {2}} $ be drawn from the point $P $, touches it at the point $Q$. Prove that the triangle $APQ$ is isosceles.
Let $P$ be a convex $2006$-gon. Drawn $1003$ diagonals connecting opposite vertices and $1003$ lines connecting the midpoints of opposite sides, all these $2006$ lines intersect at one point. Prove that the opposite sides $P$ are parallel and equal.
In the right triangle $ABC$ $ (\angle A = 90^\circ)$ on the side $AC$ take the point $D$. The point $E $ is a mirror image of the point $A$ wrt $BD$, and the line $CE$ intersects the perpendicular drawn from the point $D$ on $CB$, at the point $F$. Prove that the lines $AF, DE, CB$ intersect at one point.
In an acute triangle $ABC$, $M$ is a point on the segment $AC$ and $N$ is a point on the extension of the segment $AC$ such that $MN = AC$. Points $D, E$ are the feet of the perpendiculars drawn from points $M, N$ on the lines $BC, AB $ respectively. Prove that the orthocenter of triangle $ABC$ lies on the circle circumscribed around triangle $BED$.
Let $PA$ and $PB$ be tangents to the circle $w$ from the point $P$. Let $M, N$ be the midpoints of the segments $AP, AB$ respectively. The extension of $MN$ intersects $w$ at the point $C$, where $N$ lies between $C$ and $M$. $PC$ intersects $w$ at point $D$, and the extension of $ND$ intersects $PB$ at point $Q$. Prove that $MNPQ$ is a rhombus.
Let $O$ be the center of the circle circumscribed around $\Delta ABC$, and the lines $AO$ and $BC$ intersect at the point $D$. On the line $BO$ such point $S$ is chosen that $AB \parallel DS $. The lines $AS$ and $BC$ intersect at the point $T$. Prove that if the points $O, D, S, T$ lie on one circle, then $\Delta ABC$ is isosceles
Let the quadrilateral $ABCD$ be inscribed in a circle with center $O$, with obtuse angles $\angle B$ and $\angle C$. Let $E$ be the point of intersection of the lines $AB$ and $CD$. $P$ and $R$ are the bases of the perpendiculars drawn from the point $E$ on $BC$ and $AD$, respectively. Let $Q$ be the point of intersection of the lines $EP$ and $AD$. Let $S$ be the point of intersection of the lines $ER$ and $BC$. Denote by $K$ the midpoint of the segment $QS$. Prove that the points $E, K, O$ lie on the same line.
In the quadrilateral $PQRS$, the points $A, B, C, D$ are the midpoints of the sides $PQ$, $QR$, $RS$ ,$SP$ respectively, and $M$ be the midpoint of $CD$. Let $H$ be a point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90 {}^ \circ$.
Given a circle $\Gamma$ with center at point $O$ and point $A$ outside the circle. A line passing through $A$ intersects the circle $\Gamma$ at the points $X$ and $Y$. The point $Z$ is symmetric to the point $X$ with respect to the line $OA$. Prove that the point of intersection of the lines $OA$ and $ZY$ does not depend on the choice of that line.
A polyline is drawn inside the square $50 \times 50$, and the distance from any point inside the square to the polyline is not more than $1$. Prove that the length of the broken line is not less than $1248$.
A semicircle with diameter $AB$ and center at point $S$ is specified. The points $C$ and $D$ are marked on this semicircle (the point $D$ lies on the arc $BC$) so that $\angle CSD = 90^\circ$ . Let $E$ be the point of intersection of the lines $AC$ and $BD$, $F$ be the point of intersection of the lines $AD$ and $BC$. Prove that $EF = AB$.
The diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ intersect at the point $E$. The points $M$ and $N$ are the midpoints of the segments $AE$ and $CD$, respectively. It is known that $BD$ is a bisector $\angle ABC$. Prove that the quadrilateral $ABCD$ is cyclic if and only if the quadrilateral $MBCN$ is cyclic .
The inscribed circle of the triangle $ABC$ ($AC \ne BC$) touches the sides $AB$, $BC$ and $CA$ at the points $P$, $Q$ and $R$ respectively. Denote by $G$ the point of intersection of the medians and by $I$ the center of the inscribed circle of the triangle $ABC$. It turned out that $GI \bot AB$. Prove that:
a) the points $R$, $G$ and $Q$ lie on the same line,
b) $CR = AB$.
In the inscribed pentagon $ABCDE$, side $BC = 7$. The diagonals $EC$ and $AC$ intersect the diagonal $BD$ at the points $L$ and $K$, respectively. It is known that the points $A, K, L, E$ lie on the same circle $\Gamma$. From the point $C$ the tangent $CO$ is drawn to the circle $\Gamma$, where $O$ is the point of contact. Find the length of $CO$.
In the acute-angled triangle $ABC$, the altitude $CH$ and the median $BM$, are drawn, which intersect at point $T$. It turned out that $\angle MCH = \angle MBC$ and $CH = BM$. Find the angles of triangle $ABC$.
The chord $PQ$ with point $R$ as the midpoint is drawn in a circle with diameter $AB$. Two perpendiculars $PS$ and $QT$ are drawn on this diameter. Prove that the triangle $RST$ is equilateral if and only if $2PQ = AB$.
Two circles $\gamma_1$ and $\gamma_2$ with centers at points $O_1$ and $O_2$ and radii $4$ and $9$ respectively touch externally at point $B$. A common external tangent to these circles touches the larger circle at point N and intersects the common interior tangent at the point $K$. Find the radius of the circle inscribed in the quadrilateral $BKNO_2$.
Let $ABCD$ be a convex quadrilateral inscribed in a circle with center at point $O$ and radius $R$ (through $(O, R)$ here and in what follows we denote a circle with center at the point $O$ and radius $R$). Consider four circles $\gamma_A = (A, R)$, $\gamma_B = (B, R)$, $ \gamma_C = (C, R)$, $\gamma_D = (D, R)$, and denote thus their points of intersection are different from the point $O$: $K \in \gamma_A \cap \gamma_B$, $L \in \gamma_B \cap \gamma_C$, $M \in \gamma_C \cap \gamma_D$, $N \in \gamma_D \cap \gamma_A$. Prove that $KLMN$ is a parallelogram.
Let $O$ be the center of the circumcircle of the triangle $ABC$, the points $A_1, B_1, C_1$ be the midpoints of the corresponding sides, points $A_2, B_2, C_2$ are defined conditions $\overrightarrow{OA_2} = \lambda \cdot \overrightarrow{OA_1}$,$\overrightarrow{OB_2} = \lambda \cdot \overrightarrow{OB_1}$, $\overrightarrow{OC_2} = \lambda \cdot \overrightarrow{OC_1}$ for $\lambda> 0$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point.
In the acute-angled triangle $ABC$, the altitudes $BK$ and $AL$ are drawn . Let $H$ be the point of intersection of altitudes , and let point $M$ be the midpoint of the side $AB$. Prove that the bisector of $\angle KML$ passes through the middle of the segment $CH$.
Let $ABC$ be a triangle in which $AB> AC$, $AM$ and $AK$ are the median and angle bisector of this triangle. Point $L$ is on the line $AM$ such that $KL \parallel AC$. Prove that $CL \perp AK$.
A circle inscribed in an acute-angled right triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$ and$ F$, respectively. The point $H$ is selected on the segment $EF$ such that $DH \perp EF$. Prove that if $AH \perp BC$, then $H$ is the orthocenter of $\vartriangle ABC$.
Given an acute triangle $ABC$ with angle $\angle ACB = 60^o$. Points $A_1, B_1$ are selected on sides $BC$ and $AC$, respectively. Point $D$ is the second point of intersection of the circles circumscribed around $\vartriangle BCB_1$ and $\vartriangle ACA_1$, different from point $C$. Prove that $D$ lies on the side $AB$ if and only if $\frac{CB_1}{CB} + \frac{CA_1}{CA} = 1$.
Given a rectangle $ABCD$ with sides $AB = a$ and $BC = b$, O is the point of intersection of the diagonals. On ray $BA$ beyond point $A$ lies the segment $AE = AO$, and on ray $DB$ beyond point $B$ lies the segment $BZ = BO$. It is known that the triangle $EZC$ is equilateral. Prove that:
a) $AZ = EO$
b) $EO \perp ZD$.
In the triangle $ABC$, in which $AB\ne AC$, the altitude $AD$ is drawn. Points $E$ and $F$ are the midpoints , respectively of segments$ AD$ and $BC$. Point $G$ is the foot of the perpendicular drawn from point $B$ on the line $AF$. Prove that the line $EF$ touches the circle circumcscribed around $\vartriangle CGF$, at point $F$.
In a rectangle $ABCD$ with sides $AB> BC$, the perpendicular bisector of the diagonal $AC$ intersects the side $DC$ at point $E$. A circle with center at point $E$ and radius AE intersects segment $AB$ for second time at point $F$. The point $G$ is the foot of the perpendicular drawn from the point $C$ on the segment $EF$. Prove that the point $G$ lies on the diagonal $BD$.
The bisectors of the angles $A, B$ and $C$ of the triangle $ABC$ intersect the circle circumscribed around this triangle S_1 (O, R) at points $A_2, B_2, C_2$, respectively. Tangents to the circle $S_1$ at points $A_2, B_2, C_2$ intersect between at points $A_3, B_3, C_3$ (points $A$ and $A_3$ lie on one side of the line $BC$, similarly for others points). Let the circle $S_2 (I, r)$ be inscribed in the triangle $ABC$ touches its sides at points $A_1, B_1, C_1$, respectively (point $A_1 \in BC$, similarly for other points). Prove that lines $A_1A_2$, $B_1B_2$, $C_1C_2$, $AA_3$, $BB_3$, $CC_3$ intersect at one point.
Through the point $L$, the midpoint of the side BC of the triangle $ABC$, in which $AC <AB$, draw a line $\ell$ parallel to the bisector $AV$ of the angle $BAC$. The line $\ell$ intersects the lines $AB$ and $AC$ at the points $X$ and $Y$, respectively. On the point $Z$ is marked by the point $Z$, for which the equality $XY = Y Z$ holds. The lines $BY$ and $CZ$ intersect at point $D$. Prove that the bisector of $\angle BDC$ is parallel to the line $.
Let $G$ and $O$, respectively, be the point of intersection of the medians and the center of the circle circumscribed around $\vartriangle ABC$ The perpendicular bisectors of the segments $GA, GB$, and $GC$ intersect at points $A_1, B_1$, and $C_1$. Prove, that $O$ is the point of intersection of the medians of the triangle $A_1B_1C_1$.
The circle ${{\gamma} _ {1}}$ is inscribed in triangle $ABC$ and touches the sides $AB$ and $AC$ at points $D$ and $E$, respectively. The circle ${{\gamma} _ {2}} $ is inscribed in triangle $ADE$. Prove that the center of the circle ${{\gamma} _ {2}}$ is located on the circle ${{\gamma} _ {1}}$.
The circle ${{\gamma} _ {1}}$ is inscribed in triangle $ABC$ and touches the sides $AB$ and $AC$ at points $D$ and $E$, respectively. The circle ${{\gamma} _ {2}}$ is inscribed in triangle $ADE$ and touches the lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $M$ and $N$ be the points of intersection of the circles ${{\gamma} _ {1}} $ and ${{\gamma} _ {2}} $. Denote the radii of the circles ${{\gamma} _ {1}} $ and ${{\gamma} _ {2}}$ by ${{r} _ {1}} $ and ${ {r} _ {2}} $ respectively. Prove that the points $M, N, P$ and $Q$ form a rectangle if and only if the equality holds: $2 {{r} _ {1}} = 3 {{r} _ {2}} $.
The point $P$ is chosen inside the triangle $ABC$ so that $\angle CAP = \angle BCP$. Let ${B} '= BP \cap AC$, ${C}' = CP \cap AB $, and $Q$ be the point of intersection of the line $AP$ with the circle circumscribed around $\Delta ABC $, different from $A$. Let also $R = {B} 'Q \cap CP $ and $S = {B}' Q \cap \ell $, where $\ell$ is a line passing through the point $P$ and parallel to $AC$. The point $T = {B} '{C}' \cap QB$ lies on the other side of the line $AB$ than the point $C$. The circle circumscribed around $\Delta P {B} '{C}' $, the second intersects the line $AP$ at a point lying inside the triangle $ABC$. Prove that $\angle BAT = \angle B {B} 'Q \Leftrightarrow SQ = R {B}' $, that is, the angles $\angle BAT$ and $\angle B {B} 'Q$ are equal if and only if the segments $SQ$ and $R {B}' $ are equal. .
On the sides of the scalene acute $\Delta ABC$, choose the points $X \in AB$ and $Y \in AC$ so that $BX = CY $. Prove that, regardless of the choice of points $X$ and $Y$, the circle circumscribed around $\Delta AXY$ passes through some fixed point other than $A$.
Through the point $P$, which is located inside the triangle $ABC$, the Cevians $A {{A} _ {1}}$, $B {{B} _ {1}} $ and $C {{C} _ {1}} $. let ${{P} _ {1}} $ be some point inside $\Delta {{A} _ {1}} {{B} _ {1}} {{C} _ {1}}$ . Let ${{A} _ {2}} = {{A} _ {1}} {{P} _ {1}} \cap {{B} _ {1}} {{C} _ {1} } $, ${{B} _ {2}} = {{B} _ {1}} {{P} _ {1}} \cap {{A} _ {1}} {{C} _ {1}}$ and ${{C} _ {2}} = {{C} _ {1}} {{P} _ {1}} \cap {{A} _ {1}} {{ B} _ {1}} $. Prove that the lines $A {{A} _ {2}}$, $B {{B} _ {2}}$ and $C {{C} _ {2}} $ intersect at one point.
Let $\Delta ABC$ be an acute triangle, ${B} '$ and${C}' $ be points symmetric to the points $B$ and $C$ wrt the lines $AC$ and $AB$, respectively, and $P$ be the intersection point of the circles circumscribed around $\Delta AB {B} '$ and $\Delta AC {C}' $, other than $A$. Prove that the center of the circle circumscribed around $\Delta ABC$ lies on the line $AP$.
On the side $BC$ of the parallelogram $ABCD$ with acute angle $A$, the point $T$ is chosen so that $\angle ATD$ is acute and $\angle ADT <\angle BAD$. Prove that the centers of the circumcircles of the triangles $ABT, ADT$ and $CDT$ form a triangle whose orthocenter lies on the segment $AD$.
The angle bisectors $B {{B} _ {0}} $ and $C {{C} _ {0}}$ are drawn in the triangle $ABC$. The extensions of these bisectors intersect the circle circumscribed around $\Delta ABC$ at the points ${{B} _ {1}} $and ${{C} _ {1}} $, respectively. The lines ${{B} _ {0}} {{C} _ {0}}$and ${{B} _ {1}} {{C} _ {1}} $ intersect at point $X$. Also let $I = B {{B} _ {0}} \cap C {{C} _ {0}}$. Show that $X \ne I $, and prove that $XI \parallel BC$.
A pentagon $ABCDE$ inscribed in a circle with sides $AB = BC$ and $CD = DE$ is given. The diagonals $AD$ and $BE$ intersect at the point $P$, and the diagonal $BD$ intersects the segments $CA$ and $CE$ at the points $Q$ and $T$ respectively. Prove that $\Delta PQT $ is isosceles.
Given an isosceles triangle $OCB$ with vertex $O$, circle $w = S (O, \, \, OB) $. Tangents to this circle at the points $B$ and $C$ intersect at the point $A$. Consider the circle ${{w} _ {1}}$, located inside $\Delta ABC$, which touches the circle $w$ and the side $AC$ at the point $H$. Circle ${{w} _ {2}}$ is also inside $\Delta ABC$ and touches circle $w $ and circle ${{w} _ {1}}$ at point $J$, and the sides $AB$ at the point $K$. Prove that the bisector of $\angle KJH$ passes through the incenter of $\Delta OCB$.
A hexagon is circumscribed around the circle, in which the opposite sides are parallel in pairs. Prove that the opposite sides are equal in pairs.
Let $ABCD$ be a trapezoid with bases $AB> CD$. The points $E$ and $F$ are selected on the segments $AB$ and $CD$ respectively so that $\frac {AE} {EB} = \frac {DF} {FC}$. Let $K$ and $L$ be two points on the segment such that $ \angle AKB = \angle DCB$ and $\angle CLD = \angle CBA$. Prove that the points $K$, $L$, $B$ and $C$ are concyclic.
In the triangle $ABC$ draw the medians $AM, \, \, BN, \, \, CP$, the extensions of which intersects the circumscribed circle at the points $D, \, \, E, \, \, F$ respectively, $G$ is the point of intersection of the medians. Prove that the inequality holds:
$$\sqrt {3} \left (\frac {1} {a} + \frac {1} {b} + \frac {1} {c} \right) \ge \frac {1} {GD} + \frac {1} {GE} + \frac {1} {GF} \ge \frac {3} {R}.$$where $a, \, \, b, \, \, c$ are the sides of $\Delta ABC$, and $R$ is the radius of the circumscribed circle.
In the trapezoid $ABCD$ the diagonal $AC$is equal to the side $CD$. The line that is symmetric to the line $BD$ relative to $AD$ intersects with the line $AC$ at the point $E$. Prove that the line $AB$ divides the segment $DE $ in half.
In the convex quadrilateral $ABCD$ on the side $AB$ the points $E$ and $F$ are selected, and on the side $CD$ the point $G$ is selected such that the quadrilaterals $ABCG$, $AFCD$ and $EFCG$ are cyclic. Prove that $AE = FB$ if and only if $AB\parallel CD$.
In the triangle $ABC$, the equality $BC = 2AC$ holds. On the side $BC$ the point $D$ is selected, such that $\angle CAD = \angle CBA$. The point $F$ is selected on the ray $AC$ so that the point $C$ lies on the segment $AF$. The line $AD$ intersects the bisector of $\angle FCB$ at the point $E $. Prove that $AE = AB$.
The diagonals of the quadrilateral $ABCD$ intersect at the point $O$. Denote the feet of the perpendiculars drawn on the sides $AB, \, \, BC, \, \, CD, \, \, DA$ by $P, \, \, Q, \, \, R, \, \, S$ respectively (each point falls exactly on the side of the quadrilateral). Prove that equality
$$PA \cdot AB + RC \cdot CD = \frac {1} {2} (A {{D} ^ {2}} + B {{C} ^ {2}}) $$is true if and only if is true the equality:
$$QB \cdot BC + SD \cdot DA = \frac {1} {2} (A {{B} ^ {2}} + C {{D} ^ {2}}).$$
Inside the circle $\Gamma$ of unit radius are several smaller circles whose total length is greater than or equal to $\pi$ and none of them contains the center of the unit circle inside. Prove that there exists a concentric circle to $\Gamma$ that intersects at least two small circles.
In the isosceles $\Delta ABC$ the angle at the base $BC$ is equal to $80 {} ^ \circ $. On the side $AB$ the point $D$ is chosen such that $AD = BC$, and on the ray $CB$ the point $E$ is chosen such that $AC = EC $. Find the angle $EDC$ .
Given a circle $ w $ and points $ A, \, \, B $ on a line that does not intersect the circle $ w $. An arbitrary point $ {{X} _ {0}} $ is selected on the circle and the following sequences of points $ ({{Y} _ {n}}) $ and $ ({{X} _ {n}}) $ are constructed: $ {{Y} _ {n}} $ is the second intersection point of of the line $ A {{X} _ {n}} $ with the circle $ w $, $ n \ge 0 $, and $ {{X} _ { n + 1}} $ is the second intersection point of of the line $B {{Y} _ {n}}$ with the circle $ w $, $ n \ge 0 $. Prove that if for some point $ {{X} _ {0}} $ and for some natural $ k $ the point $ {{X} _ {k}} = {{X} _ {0}} $, then $ {{X} _ {k}} = {{X} _ {0}} $ by randomly selecting the point $ {{X} _ {0}} $.
In the acute-angled triangle $ ABC $ the altitude $ CH $ is drawn, $ O $ is the center of the circumcircle, $ T $ is the foot of the perpendicular drawn from the vertex $ C $ on the line $ AO $. Prove that $ TH $ bisects the side $ BC $.
In the acute-angled triangle $ ABC $, the points $D, \, \, E, \, \, F$ are the midpoints of the sides $BC$, $CA$ and $AB$, respectively. Construct a circle $w$ with center at the orthocenter $\Delta ABC$ and such that $\Delta ABC$ is inside this circle. Let the rays $EF$, $FD$ and $DE$ intersect the circle $w$ at the points $P$,$Q$ and $R$, respectively. Prove that $AP = BQ = CR$.
Prove that in an non isosceles triangle $ ABC $ the common tangent of the Nine Points Circle and the inscribed circle is parallel to the Euler line if and only if the angles of the triangle form an arithmetic progression.
The circle $w$ inscribed in $\Delta ABC$ touches its sides $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$ respectively. A line passing through $A$ intersects the arc $EF$ , not containing the point $D$, at the point $T$. A tangent to the circle $w$ at the point $T$ intersects $EF$ at the point $P$, and the line passing through the point $P$ parallel to $AB$ intersects $AT$ at point $H$. Prove that $\angle HEF = 90 {} ^ \circ$.
Let $ABCD$ be a convex quadrilateral. Denote by ${{\gamma} _ {AB}}$, ${{\gamma} _ {BC}}$, ${{\gamma} _ {CD}}$, ${ {\gamma} _ {DA}}$ circles constructed on the sides $ AB $, $ BC $, $ CD $, $ DA $ respectively on diameters. It is known that the circles ${{\gamma} _ {AB}} $ and ${{\gamma} _ {CD}}$ touch each other, as well as the circles ${{\gamma} _ {BC }}$ and ${{\gamma} _ {DA}}$ touch each other. Prove that $ABCD$ is a rhombus.
The point $I$ is the incenter of the right triangle $ABC$, the ray $AI$ crosses the circumscribed circle $\Delta ABC$ for the second time at the point $D$. The circle passing through the points $C$, $D$, $I$, intersects the ray $BI$ again at $K$. Prove that $BK = CK$.
Let $ PQ $ be the diameter of the semicircle $ H $. The circle $ w $ internally touches $ H $, and also touches $ PQ $ at point C. Let the points $ A \in H $ and $ B \in PQ $ be such that $ AB \perp PQ $ and touches the circle $ w $. Prove that $ AC $ is the bisector of $ \angle PAB $.
In the triangle $ABC$, $AB \ ne AC$, the point $M$ is the midpoint of the arc $BC$ of the circumcircle of $\Delta ABC$, which contains the point $A $. The circle inscribed in $\Delta ABC$ centered at the point $I$ touches the side $BC$ at the point $D$. A line that is parallel to $AI$ and passes through the point $D$ intersects the inscribed circle for second time at the point $P$. Prove that the lines $AP$ and $IM$ intersect at a point on the circumscribed circle.
The acute $ \Delta ABC $ is given, in which the altitude $AD$ and $BE$, which intersect at the point $H$, are drawn. The line passing through $H$ intersects the sides $BC$ and $AC$ at the points $P$ and $Q$, respectively. The points $K \in BE$ and $L \in AD$ are such that $PK \perp BE$ and $QL \perp AD$. Prove that $DK \parallel EL$.
The triangle $ABC$ is inscribed in the circle $w$, the points $H, \, \, I, \, \, O$ are its orthocenter, incenter and center of the circumscribed circle. The line $CI $ intersects the circle $w$ for the second time at the point $D$. Find the angles $\Delta ABC$ if $AB = ID$ and $AH = OH$.
The point $P$ lies inside the triangle $ABC$ and satisfies the condition $\angle ABP = \angle PCA$, the point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP $
Inscribed in a circle, quadrilateral $ABCD$ satisfies the conditions $AD = BD$. Let $M$ be the intersection point of the diagonals of the quadrilateral, $I$ be the center of the circle inscribed in $\Delta BCM $, $ N$ be the intersection point of the line $AC$ and the circumscribed circle of $\Delta BMI$, other than $M$. Prove that $AN \cdot NC = CD \cdot BN$.
There are two circles on the plane ${{w} _ {1}}$ and ${{w} _ {2}} $ with centers ${{O} _ {1}} $ and $ {{O} _ {2}}$ respectively touch externally at the point $M $, with the radius of the circle ${{w} _ {2}} $ larger than the radius of the circle ${{w} _ {1}} $. Consider a point $ A \in {{w} _ {2}} $ such that the points ${{O} _ {1}} $, ${{O} _ {2}}$ and $A$ are not collinear. $AB $ and $AC$ are tangents to the circle ${{w} _ {1}}$ ($B$ and $C$ are points of contact). Lines $MB$ and $MC $ intersect the second circle ${{w} _ {2}} $ at points $E$ and $F$, respectively. The point of intersection $EF$ and tangent at the point $A $ to the circle ${{w} _ {2}} $ is denoted by $D $. Prove that the point $D$ lies on a fixed line when the point $A $ moves in a circle ${{w} _ {2}} $ such that the points ${{O} _ { 1}} $, ${{O} _ {2}}$ and $A $ are not collinear.
Let $ABC$ be an acute-angled triangle with $AC> BC $. Let $BN$ be its altitude , $CP$ be the median, $H$ be the orthocenter. The circumscribed circles of triangles $ABC$ and $CHN $ intersect at points $C$ and $D$. Prove that the points $B, \, \, D, \, \, N, \, \, P$ lie on the same circle.
Given the circle $ k $ and the point $A$ outside this circle. Find the locus of the orthocenters of triangles $ABC$, where $BC$ is any diameter of the circle $k$.
On the side $AB$ of the acute triangle $ABC$ as on the diameter a circle $k$ was constructed. A circle tangent to the bisector of $\angle CAB$ at the point $A$ passing through the point $C$ intersects the circle $k$ at the point $P \ne A$. Another similar circle touches the bisector of $\angle CBA$ at the point $B$ passing through the point $C$ intersects the circle $k$ at the point $Q \ne A$. Prove that the lines $AQ$ and $BP$ intersect at a point lying on the bisector of $\angle ACB$.
In the triangle $ ABC $ the equality $ AC = 2AB $ holds, and $ AD $ is its bisector. Let $ F $ be the point of intersection of the line passing through $ C $ parallel to $ AB $ and the perpendicular to the line $ AD $ constructed at the point $ A $. Prove that the line $ FD $ intersects the segment $ AC $ in its midpoint.
Given a convex quadrilateral $ KLMN $, in which $ \angle NKL = {{90} ^ {\circ}} $. Let $ P $ be the midpoint of the segment $ LM $. It turns out that $ \angle KNL = \angle MKP $. Prove that $ \angle KNM = \angle LKP $.
Given a convex quadrilateral $ PRST $, in which $ \angle TPR = {{90} ^ {\circ}} $. Let $ Q $ be the midpoint of the side $ RS $. It turns out that $\angle PTS = \angle RPQ $. Prove that $ \angle PTR = \angle SPQ $
In the convex quadrilateral $ ABCD $ on the side $ AB $ the points $ P $ and $ Q $ are selected (the point $ P $ lies between $ A $ and $ Q $), and on the side $ CD $ the points $ R $ and $T $ are selected (point $ R $ lies between $ C $ and $ T $). It turned out that $ AP = PT = TD $ and $ QB = BC = CR $. You can also circumscribe a circle around the quadrilateral $ BCTP $. Prove that a circle can also be circumscribed around the quadrilateral $ ADRQ $.
A right triangle $ABC$ with a right angle $C$ is given. Using the legs of this triangle as bases are built isosceles triangles $ACD$ and $ECB$ outside the triangle $ABC$ , such that $\angle ABC = \angle ADC$ and $\angle BAC= \angle BEC$. Let $M$ be the midpoint of the side $AB$. Prove that $DM+ME$ is equal to the perimeter of $ABC$.
In the rectangle $ABCD$ on the diagonal $AC$, a point $K$ is selected such that $BC=CK$. On the side $BC$, a point $M$ is selected such that $ KM = MC $. Prove that $AK+BM=CM$.
In a right isosceles triangle $ABC$ , on the hypotenuse $CB$, the point chosen $M$ is such that $\angle AMB=75^o$. Inside the $\vartriangle ABC$ , a point $F$ lies on the bisector of $\angle CAM$, such that $BF=AB$. Prove that:
a) $AM \perp BF$
b) $\vartriangle CFM$ is isosceles.
The point $O$ is the center of the circle circumscribed around the acute-angled triangle $ABC$. Circles $c_1$ and $c_2$ are circumscribed around triangles $ABO$ and $ACO$, respectively. The points $P$ and $Q$ are chosen on $c_1$ and $c_2$ respectively so that $OP$ is the diameter of $c_1$ and $OQ$ is the diameter of $c_2$. Let T be the intersection point of the tangent to the circle at the point $P$ and the tangent to the circle at the point $Q$. Let $D$ be the second intersection point of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ belong to the same line.
The line $l $ intersects the right branch of the hyperbola $y = \frac {1} {x}, \, \, x> 0$ at the points $A$ and $B$. Lines ${{l} _ {1}}, \, \, {{l} _ {2}}$, which are parallel to the line $l $, intersect the left branch of the same hyperbola $(x <0 )$ at points $E,F$ and $C,D$, respectively. The segment $AD$ intersects the line $l_1$ at the point $G$, and the segment $BC$ intersects the line ${{l} _ {1}} $ at the point $H$. Prove that $GE=HF$.
Let $ABC$ be an acute-angled triangle with altitudes $BD$ and $CE$. Points $S$ and $T$ are symmetric points of $E$ wrt lines $AC$ and $BC$ respectively. The circle circumscribed around $\vartriangle CST$ has center $O$ and intersects the line $AC$ for second time at point $X$. Prove that $XO \perp DE$.
Given an acute $\Delta ABC$. Denote the following points: $D$ is the foot of the perpendicular drawn from the vertex $A$ to the side $BC$, $M$ is the midpoint of $BC$, $H$ is the orthocenter $\Delta ABC$, $E$ is the intersection point of the circumscribed circle $\Gamma$ of $\Delta ABC$ and the ray $MH$, $F$ is the intersection point of (other than $E$) line $ED$ and the circle $\Gamma$. Prove that the equality holds:$$\frac {BF} {CF} = \frac {AB} {AC} .$$
In the acute triangle $ABC$, the altitude $BH$, the midline $DE || BC$, $D \in AB$ are drawn. The point $F$ is symmetric to the point $H$ wrt $DE$. Prove that the line $BF$ passes through the center of the circumscribed circle $\Delta ABC $.
Points $M, \, \, N, \, \, K$ are midpoints of sides $BC$, $AC$ and $AB$ respectively of $\Delta ABC$. Let ${{w} _ {B}} $ and ${{w} _ {C}}$be two semicircles built using the sides $AC$ and $AB$, respectively, as the diameters, to the outside of $\Delta ABC$. Let $MK$ and $MN$intersect the semicircles ${{w} _ {C}} $ and ${{w} _ {B}} $ at the points $X$ and $Y $ respectively. The point $Z$ is the point of intersection of the tangents ${{w} _ {C}} $ and ${{w} _ {B}} $ at the points $X $ and $Y$ respectively. Prove that $AZ \perp BC$.
In the isosceles triangle $ABC$, point $P$ is marked on the base of $AC$, and point $Q$ is marked on the side $BC$, such that $AB = CP$ and $AP = BP = PQ$. Prove that $AQ$ is the bisector of the angle $BAC$.
In a convex quadrilateral $ABCD$ with right angles at vertices $B$ and $D$ on the extension of side $AB$ beyond point $A$, a point $P$ is chosen such that $\angle BCP = \angle BAD$. Point $Q$ is symmetric to point $D$ wrt point $B$. Prove that $\angle BAC = \angle BQP$.
In an acute-angled triangle $ABC$, the altitudes intersect at point $H$ and the angle bisectors intersect at point $I$. The circle circumscribed around triangle $IBC$ intersects segment $AB$ at point $T$. Point $Q$ is the foot of the perpendicular drawn from point $H$ on line $IA$. Point $P$ is such that $Q$ is the midpoint of the segment $PT$. Prove that points $B, H$ and $P$ are collinear.
Denote by$c (O, \, \, R)$- the circumcircle of the acute triangle $ABC$, where $O$ and $R$ are the center and radius of the circle $c$ . Let $F$ be a point on the side $AB$ such that $AF <\frac {1} {2} AB$. The circle ${{c} _ {1}} (F, \, \, FA) $ intersects the line $OA$ at the point $A '$ and the circle $c$ at the point $K$. Prove that the quadrilateral $BKFA '$ is inscribed in a circle passing through the point $O$.
The acute triangle $ABC$ has a center at the point $I$. A line perpendicular to the line $BI$ at the point $I$ intersects the sides $BA$ and $BC$ at the points $E$ and $D$, respectively. The points $P$ and $Q$ are the incenters of $\Delta ABI$ and $\Delta CBI$. Prove that if the points $D, \, \, E, \, \, P, \, \, Q$ are cyclic, then $AB = BC$.
The circle inscribed in the triangle $ABC$ touches the sides $BC$, $AC$ and $AB$ at the points $D$, $E$ and $F$, respectively. The line passing through the point $F$ perpendicular to $FE$ intersects with the line $ED$ at the point $P$. Similarly, a line passing through the point $D$ perpendicular to $DE$ intersects with the line $EF$ at the point $Q$. Prove that the point $B$ is the midpoint of the segment $PQ $.
Circle $k$ is an exscribed circle of triangle $ABC$ tangent to side $BC$ at point $K$ and to extensions of sides $AB$ and $AC$ at points $L$ and $M$ respectively. A circle with diameter $BC$ intersects the segment $LM$ at the points $P, \, \, Q$ (the point $P$ belongs to the segment $LQ$). Prove that $BP$ and $CQ$ intersect at the center of the circle $k$.
In an isosceles trapezoid $ABCD$ the point $O$ is the midpoint of the base $AD$. A circle centered at the point $O$ and radius $BO$ touches the line $AB$. Let the segment $AC$ intersect this circle at the point $K \ne C$, and let the point $M$ be such that $ABCM$ is a parallelogram. The circumscribed circle $\Delta CDM$ intersects the segment $AC $ at the point $L \ne C$. Prove that $AK = CL$.
In the right triangle $ABC$ with the hypotenuse $AB$, the angle bisector $BD$ is drawn. The point $E$ lies on the ray $CB$ such that $AD = DE$. Prove that the points $A, \, \, B, \, \, D, \, \, E$ lie on the same circle.
The perimeter of the triangle $ABC$ is equal to $8 $. The points $D$ and $E$ are selected on the sides $AB$ and $AC $, respectively, so that $DE\parallel BC$ and $DE$ touch the inscribed circle in $\Delta ABC$ . What is the largest value of the length of the segment $DE $?
In the triangle $ABC$ with the center of the inscribed circle $I$, the midpoints of the sides $BC$, $CA$ and $AB $ are denoted by ${{M} _ {a}} $ , ${{M} _ {b}}$ and ${{M} _ {c}} $, respectively, and the feet of the corresponding altitudes through ${{H} _ {a}}$, ${{H} _ {b}} $ and ${{H} _ {c}} $ respectively. Denote by ${{l} _ {b}}$ line tangent to the circumscribed circle $\Delta ABC$ at the point $B$. $l {{'} _ {b}}$ Is a symmetric image ${{l} _ {b}} $ wrt the line $BI$. Define the points ${{P} _ {b}} = {{M} _ {a}} {{M} _ {c}} \cap {{l} _ {b}}$ and ${{ Q} _ {b}} = {{H} _ {a}} {{H} _ {c}} \cap l {{'} _ {b}}$. Similarly, are defined ${{l} _ {c}} $, $l {{'} _ {c}} $, ${{P} _ {c}} $ and ${{ Q} _ {c}}$. Let $R = {{P} _ {b}} {{Q} _ {b}} \cap {{P} _ {c}} {{Q} _ {c}}$, prove that $RB = RC$.
Let $ABC$ be an acute triangle with $AB \ne AC$ and centroid at the point $G$. Point $M$ is the midpoint of side $BC$. The circle $\Gamma $ has center $G$ and radius $GM$. $N$ is the second intersection point of the circle $\Gamma$ and the line $BC$. The point $S$ is symmetric to the point $A$ wrt the point $N$. Prove that $GS \perp BC$.
On the sides $AB$ and $AC$ of the triangle $ABC$, the points $D$ and $E$ are selected, respectively, so that $DE\parallel BC$. Point $M$ is thw midpoint of side $BC$. The point $P$ is such that $DB = DP$ , $EC = EP$and the segments $AP $ and $BC$ intersect at the inner point. Prove that $\angle CPE = \angle BMD$ if $\angle BPD = \angle CME$.
Consider a circle $\Gamma$ with a diameter $AB$. The line $\ell$ is drawn outside the circle perpendicular to the diameter $AB$. The points $X, \, \, Y $ are selected on this line that does not lie on the diameter $AB$. Let the points $X ', \, \, Y' $ be selected on $\ell$ so that $AX \cap BX '\in \Gamma$ and $AY \cap BY' \in \Gamma $. Prove that the circumscribed circles of $\Delta AXY$and $\Delta AX'Y$ either intersect at the circle $\Gamma $ at a point other than $A$, or at the point $ A$ all three constructed circles touch each other.
In the triangle $ABC$ , $\angle A <\angle C$. The point $D $ lies on the extension behind the point $B$ of the segment $BC$ so that $AB = BD$. The point $E$ lies on the bisector $\angle ABC$ so that $\angle ACB = \angle BAE $. The segment $BE$ intersects the segment $AC$ at the point $F$. The point $G$ lies on the segment $AD$ in such a way that $ EG \parallel BC$. Prove that $AG = BF$.
On the sides $AB$ and $AC$ of the triangle $ABC$, arbitrary points $E$ and $F$ are selected, respectively. The circumscribed circle of $\Delta AEF$ intersects the circumscribed circle of $\Delta ABC$ for second time at the point $M$. The point $D$ is a symmetric image of the point $M$ wrt the line $EF$, the point $O$ is the center of the circumscribed circle of $\Delta ABC$. Prove that the point $D$ lies on the side $BC$ if and only if the point $O$ belongs to the circumscribed circle of $\Delta AEF$.
In the right triangle $ABC$, its legs $BC$ and $AC$ are the hypotenuses of the right isosceles triangles $BCP$ and $ACQ$, which are located outside $\Delta ABC$. The point $D$ is the vertex of a right isosceles triangle with hypotenuse $AB$, which is located on one side with the vertex $C$. Prove that $D$ lies on the line $PQ$.
The triangle $ABC$ has an orthocenter at the point $H $, different from the vertices and from the center of the circumcircle $O$ of this triangle. Let $M, \, \, N, \, \, P $ be the centers of the circumscribed circle of $\Delta HBC$ $\Delta HAC$, $\Delta HAB$, respectively. Prove that $AM, \, \, BN, \, \, CP$ and $OH$ intersect at the same point.
For the triangle $ABC $, we constructed a circle $w $, which touches the side $BC$ at the point $C$ and passes through the vertex $A$. Point $M$ is the midpoint of side $BC$. The line $AM$ intersects the circle $w$ for the second time at the point $D$, and the line $BD$ intersects the second circle $w$ at the point $E$. Prove that the circumscribed circle of $\Delta AMC$ passes through the midpoint of the segment $CE$.
Given an equilateral triangle $ABC$. On the extension of the side $AC$ beyond the point $C$ and on the extension of the side $BC$ beyond the point $C$, the points $D$ and $E$ are marked respectively, such that $AD = CE$. Prove that $BD = DE$.
Let $ABC$ be an acute isosceles triangle. The point $D$ is the midpoint of the side $BC$, the points $E$ and $F$ are the projections of the point $D$ on the sides $AB$ and $AC$, respectively. If $M$ is the midpoint of the segment $EF$ and the point $O$ is the center of the circumscribed circle of $\Delta ABC$, prove that $DM \parallel AO$.
Let the triangle $ABC$ have the center incenter $I$, and circumcircle $\Omega$ . Denote by $w$, a circle tangent to the sides $AB$and $AC$ at points $D$ and $E$, respectively, and internally tangent to the circle $\Omega$ at point $T$. The line $IT$ intersects $w $ for second time at the point $P$. Denote by $M $ and $N$ the points of intersection of the lines $PD$ and $PE$ with the line $BC$, respectively. Prove that the points $D, \, \, E, \, \, M, \, \, N $ belong to one circle and determine the center of this circle.
Let $ABC$ be an isosceles triangle with vertex at $A$ and $\angle CAB <60 ^o$. On side $AC$ select point $D$, such that $\angle DBC = \angle BAC$. Point $E$ is the intersection point of the perpendicular bisector of the segment $BD$ and the line passing through the point $A$ parallel to $BC$. Prove that $AC \parallel BE$.
Let $ABC$ be an isosceles triangle with vertex at $A$ and $\angle CAB <60 ^o$. On side $AC$ select point $S$, such that $\angle DBC = \angle BAC$. Point $E$ is the intersection point of the perpendicular bisector of the segment $BD$ and the line passing through the point $A$ parallel to $BC$. The point $F$ lies on the line $AC$ such that $A$ lies on the segment $CF$ and $AF=2AC$ . Prove that a line passing through $F$ perpendicular on $AB$, and a line that pass through $E$ perpendicular on $AC$, intersect on the line $BD$.
In an acute-angled triangle $ABC$, in which $BC<CA<AB$, the heights $AD,BE,CF$ are drawn. A line parallel to $DE$ and passing through $F$ intersects the line $BC$ at the point $M$, and the bisector of angle $\angle MFE$ intersects the line $DE$ at the point $N$. Prove that the point $F$ is the center of the circumscribed circle of $\vartriangle DMN$ if and only if $B$ is the center of the circumscribed circle $\vartriangle FMN$ .
Let $ABCD$ be a square, $k$ be a circle with center the point $B$ passing through the points $A,C$ and some point $T$ inside the square. The tangent to the circle $k$ at the point $T$ intersects the sides $CD$ and $DA$ at the points $E$ and $F$, respectively. Let $G$ and $H$ be the intersection points of the lines $BE$ and $BF$ with the segment AC, respectively. Prove that the lines $BT, EH$ and $FG$ pass through one point.
Let $k$ be a circle centered at $O$. Let $AB$ be the chord of this circle, $M$ be the midpoint of this chord. Tangents to the circle $k$ at points $A$ and $B$ intersect at point $T$. The line $\ell$ passes through the point $T$, intersects the shorter arc $AB$ at the point $C$ and the longer arc $AB$ at the point $D$ so that $BC=BM$. Prove that the center of the circumcircle of $\vartriangle ADM$ is a point symmetric to the point $O$ wrt line $AD$.
On the leg $AB$ of a right isosceles triangle ABC with a right angle at the vertex $A$ the point $D$ is selected, for which $3AD= AB$. In the half-plane given by the line $AB$ on one side with the point $C$ such point $E$ is chosen for which $\angle BDE=60^o$ and $\angle DBE= 75^o$. The point $G=BC \cap DE$ , the line passing through $G$ parallel to $AC$ intersects the line $BE$ at the point $H$. Prove that $\vartriangle CEH $ is equilateral.
From the point $P$ the tangent $PA$ to the circle $\Gamma$, $A\in \Gamma$, passes also a secant that intersects the circle $\Gamma$ at the points $B$ and $C$, such that the point $B$ lies on the segment $PC$. The point $D$ is symmetric to the point $A$ wrt the point $P$. Circles $w_1$ and $w_2$, that are circumscribed around $\vartriangle DAC$ and $\vartriangle PAB$, respectively, have a intersection point at $E \ne A$. Line $BE$ intersects the circle $w_1$ for second time at the point $F$. Prove that $CF=AB$.
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