geometry problems from OFO= OMA Foros Open, FOFO = Fake OFO , contests by argentinian math forum omaforos, with aops links in the names
collected inside aops here
Let $ABC$ be a right triangle at $B$, with $AB> BC$. Let $D$ be the foot of the altitude from $B$. Let $E$ in $AB$ such that $DE = DB$. Let $F$ be the foot of the altitude from $B$ in triangle $BEC$. Let $G$ be the intersection of $AF$ and $BC$. If $\angle DGC = 2\angle BAC$, prove that lines $BF$ and $GD$ are parallel.
You have a pentagon $ABCDE$ drawn on a blackboard. The points $M, N, P$ and $Q$ are marked, which are, respectively, the midpoints of the sides $AB$, $BC$, $CD$ and $DE$. The circumcenter of the triangle $EAB$ is also marked, which we call $O$. Everything except points $M, N, O, P$ and $Q$ is erased. Describe a procedure to reconstruct the pentagon.
Given a triangle $ABC$ we call, respectively, $D$ and $P$ the points of intersection of the bisector of $\angle BAC$ with the circumscribed circle $\Gamma$ of $ABC$ and with the side $\overline{BC}$. Let $\Omega$ be a circle that passes through points $A$ and $P$. We call $F$ and $G$ the points of intersection of $\Omega$ with line $BC$ and with circle $\Gamma$ respectively ($F \ne P$, $G \ne A$). Prove that $F, G$, and $D$ are collinear.
We have two parallelograms $ABCD$ and $AEFG$, such that $E$ is on side $BC$ and $D$ is on side $FG$. Prove that both parallelograms have the same area.
Let $\Gamma_1$ and $\Gamma_2$ be two tangent circles at point $A$, with $\Gamma_1$ inside $\Gamma_2$. A line $\ell$ that does not pass through $A$ is tangent to $\Gamma_1$ at point $B$ and intersects $\Gamma_2$ at points $C$ and $D$. Let $E \ne A$ be the second point of intersection of line $AB$ with circle $\Gamma_2$. Prove that $EC = ED$.
Let $ABC$ be a scalene acute triangle, and let $M$ be the midpoint of $BC$. Let $D$ and $E$ be the feet of the altitudes from $C$ and $B$ respectively. Let $L$ and $K$ be the midpoints of $MD$ and $ME$ respectively. Line $LK$ intersects lines $AB$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $AXMY$ is cyclic.
Let $ABCD$ be a convex quadrilateral with $\angle B =\angle D = 90^o$. Let $P$ and $Q$ be the feet of the perpendiculars drawn from $B$ on the lines $AD$ and $AC$ respectively. Let $M$ be the midpoint of $BD$. Show that the points $P, Q, M$ are collinear.
Let $ABC$ be a triangle with $B> 90^o$. It is known that there exists a point $P$ on side $AC$ such that $BP$ is perpendicular to $BC$ and $AP = BP$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$ respectively. The parallel to $AB$ drawn from $P$ intersects $DE$ at $F$. Prove that $\angle BCF = \angle ACD$.
Let $ABC$ be an acute triangle and $\omega$ its circumcircle. The altitudes $BE$ and $CF$ intersect at $H$. Let $M$ be the midpoint of $BC$ and $P$ the point of intersection of the tangents to $\omega$ by $B$ and $C$. The line $EF$ intersects the lines $PB$ and $PC$ at points $Q$ and $R$ respectively . The ray $MH$ intersects $\omega$ at $T$. Prove that the circumcircle of triangle $PQR$ and $\omega$ are tangent at point $T$.
Let $ABC$ be an isosceles triangle with $AB = AC$ and let $\Gamma$ be its circumscribed circle. On the arc $BC$ of $\Gamma$ that does not contain $A$, a point $P$ is marked, closer to $B$ than to $C$. The line perpendicular on $PC$ that passes through $A$ intersects $PC$ at $D$. Prove that $PB + PC = 2PD$.
Let $ABC$ be an acute triangle and $O$ its circumcenter. Let $\omega$ be the circumscribed circle of triangle $BOC$. Line $\ell$ is tangent to $\omega$ and cuts inside sides $AB$ and $AC$ at points $D$ and $E$ respectively. Let $A'$ be the symmetric of $A$ with respect to $\ell$. Prove that the circumscribed circles of the triangles $ABC$ and $A'DE$ are tangent..
Let $ABC$ be a triangle and let $\Omega$ be its circumscribed circle. On the arc $BC$ of $\Omega$ that does not contain $A$, a point $X$ is marked. Let $Y, Z$ be the incenters of the triangles $ABX$, $ACX$ respectively. Prove that, as $X$ varies over arc $BC$, the circumscribed circles of $XYZ$ and $\Omega$ intersect at a fixed point.
Let $ABC$ be an acute triangle and let $H$ be its orthocenter. Let $PQ$ be a segment through $H$ with $P$ on side $AB$, $Q$ on side $AC$ and such that $\angle PHB = \angle CHQ$. Finally, let $D$ be the intersection between the segment $BC$ and the bisector of the angle $\angle BAC$. Prove that $DP = DQ$.
Let $ABC$ be an acute triangle and let $O$ be its circumcenter. A circle $\omega$ through $A$ and $O$ cuts $AB$ again at $D$, $AC$ at $E$, and the circumcircle of $ABC$ at $F$. Prove that the symmetric of $F$ wrt $DE$ lies on the line $BC$.
Let $ABC$ be a scalene acute triangle and let $\ell$ be the line that passes through its orthocenter and its circumcenter. The feet of the perpendiculars on $\ell$ of $B$ and $C$ are $B_1$ and $C_1$ respectively. Rays $AB_1$ and $AC_1$ intersect the symmetric line at $\ell$ with respect to $BC$ at $B_2$ and $C_2$ respectively. Suppose that lines $BB_2$ and $CC_2$ intersect at a point $P$ on the interior of $ABC$. Prove that there exists a point $Q$ in $\ell$ such that $ABP = CBQ$ and $ACP = BCQ$.
Let $ABCD$ be a rectangle in which side $BC$ is longer than side $AB$. Point $A$ is reflected wrt the diagonal $BD$, obtaining point $E$. Knowing that $BE = EC = 404$, calculate the perimeter of the quadrilateral $BECD$.
Let $ABCD$ be a convex quadrilateral in which $\angle ABC = \angle BCD >90^o$ and $\angle CDA = 90^o$. In this quadrilateral, it is also true that $AB = 2CD$. Prove that the bisector of angle $\angle ACB$ is perpendicular to $CD$.
Let $ABCD$ be a convex quadrilateral in which $\angle ADC = 30^o$ and also $BD = AB + BC + CA$. Prove that $\angle ABD = \angle DBC$.
Let $ABCD$ be a cyclic quadrilateral. Lines $AC$ and $BD$ intersect at $R$, and lines $AB$ and $CD$ intersect at $L$. Let $M$ and $N$ be points on segments$ AB$ and $CD$, respectively, such that $\frac{AM}{MB} = \frac{CN}{ND}$. Let $P$ and $Q$ be the points of intersection of $MN$ with the diagonals $AC$ and $BD$, respectively. Prove that the circumscribed circles of the triangles $PQR$ and $LMN$ are tangent.
Let $ABC$ be a triangle with $\angle A= 90^o$. Points $D$ and $E$ are marked in such a way that $BCDE$ is a rectangle that does not overlap with triangle $ABC$ and fulfills that $BC = 2CD$. Let $M$ be the midpoint of $DE$. Prove that $\angle BAM = \angle MAC$.
In triangle $ABC$, let $L$ be the point on side $BC$ such that $AL$ is a bisector of angle $A$. Let $D$ be the midpoint of $AL$ and let $E$ be the foot of the perpendicular on $AB$ drawn from $D$. If $AC = 3AE$, prove that $LC = LE$.
Let $ABC$ be an acute triangle with $AB <AC$ and $\angle BAC = 60^o$. The perpendicular bisector of $BC$ intersects the circumscribed circle of $ABC$ at points$ E$ and $F$, with $F$ in the same half plane as $A$ wrt $BC$. Let $E'$ be the symmetric of $E$ wrt line $BC$. Let $P$ be the intersection point of the lines $CE'$ and $FA$. Prove that the circumcenter of triangle $PBC$ lies on line $AC$.
Let $ABCD$ be a convex quadrilateral with $\angle ACB = ADB = 90^o$. Lines $AC$ and $BD$ intersect at $P$, the lines $AB$ and $CD$ intersect at $Q$. Let $E$ be the symmetric of $D$ wrt $AB$. The circumscribed circles of the triangles $QAE$ and $APB$ intersect a second time at point $X$. If $M$ is the midpoint of $XP$, prove that $\angle APB = \angle AMX$.
In the triangle $ABC$, let M be the midpoint of $BC$, $N$ the midpoint of $AM$, and $D$ the point of intersection of the lines $CN$ and $AB$. Prove that if $BN=BM$, then $AD=DN$.
On a line we have four points $A, B, C, D$, in that order, so that $AB=CD$. Point $E$ is a point outside the line such that $CE=DE$.
a) Prove that if $AC=CE$, then $\angle CED=2 \angle AEB$.
b) Prove that if $\angle CED=2 \angle AEB$ , then $AC=CE$.
In the triangle $ABC$, the points $D$ and $E$ are marked on the sides $CA$ and $AB$, respectively, in such a way that the lines $BC$ and $DE$ are parallel. Line $DE$ intersects the circumcircle of $ABC$ at points $F$ and $G$, with $D$ between $F$ and $E$. Lines $FC$ and $GB$ intersect at point $P$, and the circumcircles of triangles $FEP$ and $GDP$ intersect a second time at point $Q$. Prove that the points $A, P$ and $Q$ are collinear.
Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$, $E$ and $F$ be points on the lines $BC$, $CA$ and $AB$, respectively, such that $DE\perp CO$ and $DF\perp BO$. Let $K$ be the circumcenter of triangle $AEF$. Find the locus of $K$ when $D$ varies on the line $BC$.
FOFO Anniversary 2016-22
Let $ABCDEFG$ be a convex heptagon with all sides equal. It's known that $\angle A=\angle D=168^o$ and $\angle B=\angle C=108^o$. Calculate the measure of angles $\angle E$, $\angle F$ and $\angle G$.
Let $\vartriangle ABC$ be a triangle with $AB = 10$ and $BC = 15$. Let $M$ be the midpoint of $AC$, and let $D$ be a point on the side $AC$ such that $\angle ABD = \angle MBC$. If $AD = 4$, determine the length of segment $AC$.
Let $ABC$ be a triangle, and $\omega$ be the circle with diameter $AB$. The center of $\omega$ is $O$ and $\omega$ intersects $BC$ at $Q$ and $AC$ at its midpoint $P$. Find the perimeter of $OPQ$ knowing that the perimeter of $ABC$ is $14$.
We have a parallelogram $ABCD$ of area $40$. We mark the midpoints $M$ and $N$ of $AB$ and $BC$, respectively, and $P$ the intersection of $CM$ with $DN$. Let $Q$ such that $ABQP$ is a parallelogram, and $R$ such that $ADRP $ is a parallelogram. Find the area of triangle $CQR$.
Let $ABC$ be a scalene triangle and $A_1$ its $A$-excenter. Let $A_2$ be the symmetric of $A_1$ wrt line $BC$. If $G$ and $H$ are the centroid and orthocenter of triangle $A_1BC$ respectively , prove that $GH$ is parallel to $AA_2$.
Let $ABC$ be a scalene acute triangle with circumcenter $O$. Let us call $D$ the foot of the angle bisector from $A$. Prove that the perpendicular bisector of the segment $AD$, the perpendicular to the line $BC$ through $D$, and the line $OA$ all three pass through the same point.
Let $ABCD$ be a parallelogram such that $\angle A> 90^o$, H the foot of the perpendicular from $A$ on line $BC$, and $M$ the midpoint of $AB$. Line $CM$ again intersects the circumcircle of $ABC$ at point $K$. Prove that $C, D, H, K$ are on the same circle.
Let $ABCD$ be a rhombus. On the sides $AB$ and $AD$ the points $E$ and $F$ are marked, respectively, such that $AE = DF$. Lines $BC$ and $DE$ intersect at $P$, and lines $CD$ and $BF$ intersect at $Q$. Show that $P, A$ and $Q$ lie on the same line.
Let $ABC$ be an acute triangle whose orthocenter is point $H$. The circle that passes through points $B, H$, and $C$ intersects again lines $AB$ and $AC$ at points $D$ and $E$, respectively. Let $P$ and $Q$ be the points of intersection of the segment $DE$ with $HB$ and $HC$, respectively. The points $X$ and $Y$ (different from $A$) that are on the lines $AP$ and $AQ$, respectively, are considered, so that the points $X, A, H$ and $B$ lie on the same circle and the points $Y, A, H$ and $C$ they are on the same circle. Prove that lines $XY$ and $BC$ are parallel.
Let $ABCD$ be a cyclic quadrilateral and let $E$ be the intersection of its diagonals. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively, and let $H$ be the orthocenter of triangle $BEC$. Let $P$ on line $AB$ and $Q$ on line $CD$ such that $\angle HMQ = \angle HNP = 90^o $. If $O$ is the circumcenter of $ABCD$, show that $PQ\perp OH$.
In a square $ABCD$, two points $E$ and $F$ are marked on segments $AB$ and $BC$, respectively, such that $BE = BF$. Let $H$ be the foot of the altitude of triangle $BEC$ passing through $B$. Find the measure of angle $\angle DHF$.
Let $ABCD$ be a convex cyclic quadrilateral. Let $P$ be the point of intersection of lines $AB$ and $CD$, and let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. The circumcircle of triangle $MPN$ intersects the circumcircles of triangles $ANB$ and $CMD$ a second time at points $Q$ and$ R$, respectively. Prove that $PQ = PR$.
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ be the intersection point of $BH$ with $CA$ and $F$ be the intersection point of $CH$ with $AB$. It is known that $AF=FC$. Prove that $FHEO$ is a parallelogram.
Let $ABC$ be a triangle such that $AB+AC=3BC$. The incircle $\omega$ of $ABC$ is tangent to $CA$ and $AB$ at points $K$ and $L$, respectively. We mark the points $P$ and $Q$ so that $KP$ and $LQ$ are diameters of $\omega$. Let $M$ be the midpoint of $BC$, and let $X$ and $Y$ be the intersection points of $KL$ with $BP$ and $CQ$, respectively. Prove that $MX=MY$.
FOFO Easter 2017-22
Let $ABC$ be a triangle such that $AB> AC$ and $AC> BC$, let us call $M$ the midpoint of side $AC$ and mark a point $D$ on segment $AB$ so that $BC = CD$. Knowing that $AC = BD$, prove that then it is true that $\angle BAC = 2\angle ABM$.
Let $O_1$ and $O_2$ be two circles that intersect at points $A$ and $B$. For each point C of the plane, we consider $P_1 (C)$ and $P_2 (C)$ the powers of point $C$ with respect to $O_1$ and $O_2$ respectively. Find the locus of all points $C$ such that $| P_1 (C) | = | P_2 (C) |$.
Let $ABC$ be an acute triangle with $AB <AC$ and let $O$ be the center of its circumscribed circle $\omega$. The altitude corresponding to vertex $A$ intersects segment $BC$ at $D$. Let $E$ be the second point of intersection of $AD$ with $\omega$. Let $X, Y, Z$ be the midpoints of the segments $BE$, $OD$ and $AC$ respectively. Prove that $X, Y, Z$ are collinear.
Let $ABC$ be an acute triangle and $\Gamma_A$ be the circle of diameter $BC$. Let $X_A$ and $Y_A$ be two different points on $\Gamma_A$ such that $AX_A$ and $AY_A$ are tangent to $\Gamma_A$. Similarly we define $X_B$, $X_C$, $Y_B$, and $Y_C$. Prove that there is a circle that passes through the $6$ points $X_A$, $X_B$, $X_C$, $Y_A$, $Y_B$ and $Y_C$.
Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ such that $AB + CD = AD$. Diagonals $AC$ and $BD$ intersect at point $E$. The line parallel to the bases through $E$ intersects side $AD$ at point $F$. Prove that $\angle BFC = 90^o$.
The angle bisector of $\angle BAC$ of triangle $ABC$ intersects side $BC$ and the circumscribed circle of $ABC$ at points $D$ and $E$, respectively. Let $M$ and $N$ be the midpoints of $BD$ and $CE$, respectively. The circumscribed circle of triangle $ABD$ again intersects line $AN$ at point $Q$, and the circle that passes through $A$ and is tangent to $BC$ at $D$ again intersects lines $AM$ and $AC$ at points P and R, respectively. Show that the points $B, P, Q, R$ lie on the same line.
Let $ABC$ be a triangle with $\angle BAC=40^o$ and $\angle ABC=70^o$. Let $D$ be a point on the segment $BC$ such that $AD$ is perpendicular to $BC$. We mark the point $E$ on the segment $AB$ so that $\angle ACE=10^o$. If segments $AD$ and $CE$ intersect at $F$, prove that $BC=CF$.
Let $\omega$ be the circumcircle of an acute triangle $ABC$, $D$ be the midpoint of arc $BAC$, and $I$ be the incenter of triangle $ABC$. The line $DI$ intersects $BC$ at $E$ and $\omega$ for second time at $F$. Let $P$ be at $AF$ such that $EP$ is parallel to $AI$. Prove that $PE$ bisects $\angle BPC$
COFFEE Carolina González , 9-11 May 2020
Let $ABC$ be an isosceles triangle with $AB = AC$ and $BC = 12$. Let $D$ be the midpoint of $BC$ and let $E$ be a point in $AC$ such that $DE$ is perpendicular to $AC$. The line parallel to $BC$ passing through $E$ intersects side $AB$ at point $F$.If $EC = 4$, determine the length of the segment $EF$.
Let $PQRS$ be a parallelogram, draw points$ A$ and $B$ such that $PQ = QA$, $PS = SB$, $\angle PQA = \angle PSB$ and the triangles $PQA$ and $PSB$ only share the sides $PQ$ and $PS$, respectively, with the parallelogram. Prove that $\angle RAB = \angle PAQ$ and $\angle ABR = \angle PBS$.
In a triangle $ABC$, let $K$ be a point in $AC$ such that $AK = 16$ and $KC = 20$, let $D$ be the foot of the angle bisector through $A$, and let $E$ be the point of intersection of $AD$ with $BK$. If $BD = BE = 12$, find the perimeter of triangle $ABC$.
Let $ABP$ be an isosceles triangle with $AB = AP$ and the acute angle $\angle PAB$. The line perpendicular to $BP$ is drawn through $P,$ and in this perpendicular we consider a point $C$ located on the same side as $A$ with respect to the line $BP$ and on the same side as $P$ with respect to the line $AB$. Let $D$ be such that $DA$ is parallel to $BC$ and $DC$ is parallel to $AB$, and let $M$ be the point of intersection of $PC$ and $DA$. Find $\frac{DM}{DA}$.
Let $ABC$ be a right triangle with $\angle ABC = 90^o$. Let $D$ be the symmetric of $B$ with respect to $AC$. Let a point $P$ be inside the quadrilateral $ABCD$ such that $AB = AP$. Let $E, F$, and $G$ be the feet of the perpendiculars to $BD$, $BC$, and $CD$, respectively, that pass through $P$. If $FP = 2$ and $GP = 8$, determine the length of $EP$.
Let $ABC$ be a triangle and let $D, E$ be points on the sides $AB$, $BC$, respectively, such that $2\frac{CE}{BC} = \frac{AD}{AB}$. Let $P$ be a point on side $AC$. Prove that if $DE$ is perpendicular to $PE$ then $PE$ is the bisector of the angle $\angle DPC$, and conversely, if $PE$ is the bisector of the angle $\angle DPC$ then $DE$ is perpendicular to $PE$.
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