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Thailand 2004-21 (TMO) + SHL 2008-12,-14 72p

geometry problems from Thailand Mathematical Olympiads (TMO)
with aops links in the names

in years 2004-10 only day 2 was proof based
Shortlist Geometry in aops here


2004 - 2021

A $\vartriangle ABC$ is given with $\angle A = 70^o$. The angle bisectors of $\vartriangle ABC$ intersect at $I$. Suppose that $CA + AI = BC$. Find, with proof, the value of $\angle B$.

Let $ABCD$ be a convex quadrilateral. Prove that area $(ABCD) \le \frac{AB^2 + BC^2 + CD^2 + DA^2}{4}$

A point $A$ is chosen outside a circle with diameter $BC$ so that $\vartriangle ABC$ is acute. Segments $AB$ and $AC$ intersect the circle at $D$ and $E$, respectively, and $CD$ intersects $BE$ at $F$. Line $AF$ intersects the circle again at $G$ and intersects $BC$ at $H$. Prove that $AH \cdot F H = GH^2$.

Let $O_1$ be the center of a semicircle $\omega_1$ with diameter $AB$ and let $O_2$ be the center of a circle $\omega_2$ inscribed in $\omega_1$ and which is tangent to $AB$ at $O_1$. Let $O_3$ be a point on $AB$ that is the center of a semicircle $\omega_3$ which is tangent to both $\omega_1$ and $\omega_2$. Let $P$ be the intersection of the line through $O_3$ perpendicular to $AB$ and the line through $O_2$ parallel to $AB$. Show that $P$ is the center of a circle $\Gamma$ tangent to all of $\omega_1, \omega_2$ and $\omega_3$.

From a point $P$ outside a circle, two tangents are drawn touching the circle at points $A$ and $C$. Let $B$ be a point on segment $AC$, and let segment $PB$ intersect the circle at point $Q$. The angle bisector of $\angle AQC$ intersects segment $AC$ at $R$. Show that $\frac{AB}{BC} =\left(\frac{ AR}{RC}\right)^2$

Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

Let $ABCD$ be a convex quadrilateral with the property that $MA \cdot  MC + MA  \cdot  CD = MB  \cdot  MD$, where $M$ is the intersection of the diagonals $AC$ and $BD$. The angle bisector of $\angle ACD$ is drawn intersecting ray $\overrightarrow{BA}$ at $K$. Prove that $BC = DK$ if and only if $AB \parallel CD$.

Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that$$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$

Given a $\Delta ABC$ where $\angle C = 90^{\circ}$, $D$ is a point in the interior of $\Delta ABC$ and lines $AD$ $,$ $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ at points $P$ ,$Q$ and $R$ ,respectively. Let $M$ be the midpoint of $\overline{PQ}$. Prove that, if $\angle BRP$ $ =$ $ \angle PRC$ then $MR=MC$.

Given $\Delta ABC$ and its centroid $G$, If line $AC$ is tangent to $\odot (ABG)$. Prove that $AB+BC \leq 2AC$

In $\Delta ABC$, Let the Incircle touch $\overline{BC}, \overline{CA}, \overline{AB}$ at $X,Y,Z$. Let $I_A,I_B,I_C$ be $A$,$B$,$C-$excenters, respectively. Prove that Incenter of $\Delta ABC$, orthocenter of $\Delta XYZ$ and centroid of $\Delta I_AI_BI_C$ are collinear.

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$. Let $P$ be a point on side $BC$, and let $\omega$ be the circle with diameter $CP$. Suppose that $\omega$ intersects $AC $and $AP$ again at $Q$ and $R$, respectively. Show that $CP^2 = AC \cdot CQ - AP \cdot P R$.

Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.

Let $\vartriangle ABC$ be an acute triangle, and let $P$ be the foot of altitude from $C$ to $AB$. Let $\omega$ be the circle with diameter $BC$. The tangents from $A$ to $\omega$ are drawn touching $\omega$ at $D$ and $E$. Lines $AD$ and $AE$ intersect line $BC$ at $M$ and $N$ respectively, so that $B$ lies between $M$ and $C$. Let $CP$ intersect $DE$ at $Q, ME$ intersect $ND$ at $R$, and let $QR$ intersect $BC$ at $S$. Show that $QS$ bisects $\angle DSE$

Let $\vartriangle ABC$ be a triangle with $\angle  ABC > \angle  BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?

Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .

Let $\omega$ be the incircle of $\vartriangle ABC$, $\omega$ is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$ again at $P$. Lines $AP$ and $BC$ intersect at $M$. Let $N$ be a point on segment $AC$ so that $AE = CN$. Line $BN$ intersects $\omega$ at $Q$ (closer to $B$) and intersect AM at R. Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$.

Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot  DE = BD \cdot CE$

Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:
For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle  BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.

Let $\vartriangle AB$C be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.
Let $A, B, C$ be centers of three circles that are mutually tangent externally, let $r_A, r_B, r_C$ be the radii of the circles, respectively. Let $r$ be the radius of the incircle of $\vartriangle ABC$. Prove that$$r^2  \le \frac19 (r_A^2 + r_B^2+r_C^2)$$and identify, with justification, one case where the equality is attained.

2016 Thailand MO p1
Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$

2016 Thailand MO p8
Let $\vartriangle ABC$ be an acute triangle with incenter $I$. The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$, and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of $\vartriangle XYZ$

2017 Thailand MO p2
A cyclic quadrilateral $ABCD$ has circumcenter $O$, its diagonals $AC$ and $BD$ intersect at $G$. Let $P, Q, R, S$ be the circumcenters of $\vartriangle AGB, \vartriangle BGC, \vartriangle CGD, \vartriangle DGA$ respectively. Lines $P R$ and $QS$ intersect at $M$. Show that $M$ is the midpoint of $OG$.

2017 Thailand MO p6
In an acute triangle $\vartriangle ABC$, $D$ is the foot of altitude from $A$ to $BC$. Suppose that $AD = CD$, and define $N$ as the intersection of the median $CM$ and the line $AD$. Prove that $\vartriangle ABC$ is isosceles if and only if $CN = 2AM$.

2018 Thailand MO p1
In  $\vartriangle ABC$ the incircle is tangent to $AB$ at $D$. Let $P$ be a point on $BC$ different from $B$ and $C$, and let $K$ and $L$ be incenters of  $\vartriangle ABP$ and  $\vartriangle ACP$ respectively. Suppose that the circumcircle of $\vartriangle KP L$ cuts $AP$ again at $Q$. Prove that $AD = AQ$.

2018 Thailand MO p9
In  $\vartriangle ABC$ the incircle is tangent to $AB$ at $D$. Let $P$ be a point on $BC$ different from $B$ and $C$, and let $K$ and $L$ be incenters of  $\vartriangle ABP$ and  $\vartriangle ACP$ respectively. Suppose that the circumcircle of $\vartriangle KP L$ cuts $AP$ again at $Q$. Prove that $AD = AQ$.

2019 Thailand MO p1
Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$.
(a) Show that $P$ is the circumcenter of $BDE$.
(b) Show that $A, C, D, E$ are concyclic.

2019 Thailand MO p8
Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be the circumcircle of this triangle.
Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$.
Let the circle with diameter $AI$ meets $\omega$ again at $K$.
If the line $AI$ intersects $\omega$ again at $M$, show that $K, D, M$ are collinear.

Let $\triangle ABC$ be a triangle with altitudes $AD,BE,CF$. Let the lines $AD$ and $EF$ meet at $P$, let the tangent to the circumcircle of $\triangle ADC$ at $D$ meet the line $AB$ at $X$, and let the tangent to the circumcircle of $\triangle ADB$ at $D$ meet the line $AC$ at $Y$. Prove that the line $XY$ passes through the midpoint of $DP$.

Let the incircle of an acute triangle $\triangle ABC$ touches $BC,CA$, and $AB$ at points $D,E$, and $F$, respectively. Place point $K$ on the side $AB$ so that $DF$ bisects $\angle ADK$, and place point $L$ on the side $AB$ so that $EF$ bisects $\angle BEL$.
a) Prove that $\triangle ALE\sim\triangle AEB$.
b) Prove that $FK=FL$.

Let $\triangle ABC$ be an isosceles triangle such that $AB=AC$. Let $\omega$ be a circle centered at $A$ with a radius strictly less than $AB$. Draw a tangent from $B$ to $\omega$ at $P$, and draw a tangent from $C$ to $\omega$ at $Q$. Suppose that the line $PQ$ intersects the line $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$.

Let $P$ be a point inside an acute triangle $ABC$. Let the lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at $D$ and $E$, respectively. Let the circles with diameters $BD$ and $CE$ intersect at points $S$ and $T$. Prove that if the points $A$, $S$, and $T$ are colinear, then $P$ lies on a median of $\triangle ABC$.

Geometry Shortlists 2008-12, 2014


Point $O$ is the center of the circle with chords $AB$, $CD$, $EF$ with lengths $2, 3, 4$ units, respectively, viewed from the angle at point $O$ at an angle $\alpha$, $\beta$, $\alpha + \beta$ respectively. If $\alpha + \beta$ is less than $180$ degrees, find the length of the radius of the circle.

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$ and $\angle ABC = 60^o$. Point $E$ is chosen on side $BC$ so that $BE : EC = 3 : 2$. Compute $\cos\angle CAE$.

Let $ABC$ be a triangle, let $D$ and $E$ be points on sides $AC$ and $BC$ respectively, such that $DE \parallel AB$ . Segments $AE$ and $BD$ intersect at $P$. The area of $\vartriangle ABP$ is $36$ square units and the area of $\vartriangle DEP$ is $25$ square units. Find the area of $\vartriangle ABC$.

Let $AD$ be the common chord of two equal-sized circles $O_1$ and $O_2$. Let $B$ and $C$ be points on $O_1$ and $O_2$, respectively, so that $D$ lies on the segment $BC$. Assume that $AB = 15, AD = 13$ and $BC = 18$, what is the ratio between the inradii of $\vartriangle ABD$ and $\vartriangle ACD$?

Let $ABC$ be a triangle. Angle bisectors of $\angle B$ and $\angle C$ intersect at point $O$ and intersect sides $AC$ and $AB$ at points $D$ and $E$ respectively If $OD=OE$, prove that$ \angle ABC=\angle ACB$ or $\angle ABC+\angle  ACB=120^o$.

$ABCD$ is quadrilateral with $AB=2$ , $BC=3$, $CD=7$ and $AD=6$ and $\angle ABC=90^o$. Prove that $ABCD$ is tangential and find the radius of the inscribed circle.

Triangle ABC is a triangle inscribed in a circle . Chord $CD$ bisects angle $\angle ACB$ , cut side $AB$ at point $X$, cuts the circumscribed circle at point $D$. Prove that $\frac{CX}{CA}+\frac{CX}{CB}=\frac{BA}{BD}$.

The segments $AC=6$ and $BD =4$ intersect at point $O$ at a right angle, such that $AO=2$ and $OD=3$. Lines $AB$ and $DC$ intersect at $E$ , line $EO$ intersects segment $AD$ at point $F$. Find the length of $EF$.

Let $ABC$ to be a triangle with $BC=2551$, $AC=2550$ and $AB=2549$with AD the altitude . Let the inscribed circle of triangle $BAD$ intersect $AD$ at point $E$. Let the inscribed circle of triangle $CAD$ intersect $AD$ at point $F$. Find the length of $EF$.

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.


Let $ABC$ be a triangle with the median $CD$. Point $E$ lies on side $BC$ such that $EC=\frac13 BC$ and $\angle ABC=20^o$. $AE$ intersects $CD$ at point $O$ and $\angle DAO = \angle ADO$. Find the measure of the angle $\angle ACB$.

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

Let $ABCD$ be a square with a side length of $ 1$ and $O$ be the midpoint of $AD$. A semicircle having $AD$ as diameter is drawn inside the square. Let $E$ be on the side $AB$ such that $CE$ is tangent to the circle of center $O$. Find the area of the triangle $CBE$.

Let $ABCD$ be a convex quadrilateral with side lengths $AB=BC=2$, $CD=2\sqrt3$, $DA=2\sqrt5$. Let $M$ and $N$ be the midpoints of diagonal $AC$ and $BD$ respectively and $MN=\sqrt2$. Find the area of the quadrilateral $ABCD$.

Rectangle $HOMF$ has $HO=11$ and $OM=5$ .Triangle $\vartriangle ABC$ has orthocenter $H$ and circumcenter $O$. The midpoint of side $BC$ is $M$ and the point that the altitude from $A$ meets $BC$ is $F$ . Find the length of $BC$.

In triangle $\vartriangle ABC, D$ and $E$ are midpoints of the sides $BC$ and $AC$, respectively. Lines $AD$ and $BE$ are drawn intersecting at $P$. It turns out that $\angle CAD = 15^o$ and $\angle APB = 60^o$. What is the value of $AB/BC$ ?

Let $\vartriangle ABC$ be a triangle with $AB > AC$, its incircle is tangent to $BC$ at $D$. Let $DE$ be a diameter of the incircle, and let $F$ be the intersection between line $AE$ and side $BC$. Find the ratio between the areas of $\vartriangle DEF$ and $\vartriangle ABC$ in terms of the three side lengths of$\vartriangle ABC$.

Let $O$ be the center of the circumcircle of the acute triangle $ABC$. $AO$ intersects $BC$ at point $D$. Let $S$ be the point on $BO$ such that $DS \parallel AB$. AS intersects $BC$ at point $T$. Prove that points $D,O,S,T$ lie on the same circle if and only if triangle $ABC$ is an isosceles triangle with $A$ as the vertex.

Let $ABCD$ be a convex quadrilateral with the property that $MA \cdot  MC + MA  \cdot  CD = MB  \cdot  MD$, where $M$ is the intersection of the diagonals $AC$ and $BD$. The angle bisector of $\angle ACD$ is drawn intersecting ray $\overrightarrow{BA}$ at $K$. Prove that $BC = DK$ if and only if $AB \parallel CD$.

Let $M$ be a point on the side $AC$ of the acute triangle $ABC$ .Let $N$ be a point on the extension of $AC$ beyond $C$ such that causes $MN=AC$. Let $D$ and $E$ be the projections of $M$ and $N$ on the lines $BC$ and $AB$, respectively. Prove that the orthocenter of triangle $ABC$, lies on the circumcircle of the triangle $BED$.

Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on sides $AB$ and $AC$ respectively, such that $\angle CDE=\angle ABC$. From point $D$, draw $DF\parallel BC$, that intersects $AC$ at point $F$. From point $E$, draw $EG\parallel CB$, that intersects $AB$ at point $G$. Prove that $DE^2= BG\cdot EF$

Let $\vartriangle ABC$ be an equilateral triangle, and let $M$ and $N$ be points on $AB$ and $AC$, respectively, so that $AN = BM$ and $3MB = AB$. Lines $CM$ and $BN$ intersect at $O$. Find $\angle AOB$.

Let $ABC$ be any triangle with point $D$ on side $BC$ such that $BD=\frac12 CD$. If the lengths of sides $BC$, $AC$ and $AB$ are equal to $a,b$ and $c$ respectively, then prove that$$|AD|^2 = \frac13 \left(2c^2+b^2-\frac23 a \right)$$

Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. A circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. A point $F$ on this circle is chosen so that $EF\perp  BC$. If $BC = x$, $CF = y$, and $BF = z$, find the length of $DF$ in terms of $x, y, z$.

Let $ABC$ be an acute triangle with altitude $CD$. Points $E$ and $F$ lie on line segments $AD$ and $BC$ respectively such that $\angle ECA=\angle BAF=15^o$. Let $AF$ intersect $CE$ and $CD$ at points $G$ and $H$ respectively. If triangles $AEH$ and $GFC$ are isosceles triangles with vertices at points $E$ and $G$ respectively, find the ratio between the area of triangle $EDH$ to the area of triangle $BCD$.

Let $ABCD$ be a quadrilateral with perpendicular diagonals and $\angle B=\angle D=90^o$. Let the circle of center $O$ be inscribed in the quadrilateral $ABCD$, with $M$ and $N$ the touchpoints with side $AB$ and $BC$ respectively. Let C be the center of a circle tangent to $AB$ and $AD$ at points $B$ and $D$ respectively. On extensions of $AB$ and $AD$ beyond $B$ and $D$, lie points $B'$ and $D'$ respectively, such that quadrilateral $AB'C'D'$ is similar to quadrilateral $ABCD$ and has incircle the circle of center $C$. In a quadrilateral $AB'C'D$',$N'$ is the touchpoints of it's incircle with sides $BC$. If $MN'$ is parallel to $AC'$, what is the ratio of $AB$ to $BC$, when $AB$ is longer than $BC$?

Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that$$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$

Let $ABC$ be an acute triangle with $AB>AC$. Point $D$ is lies on line segment $BC$, differs from $C$, such that $AC=AD$. Let $H$ be the orthocenter of triangle $ABC$. $A'$ and $B'$ are the feet of the altitudes drawn from points $A$ and $B$ respectively. Let the line $DH$ intersect $AC$ and $A'B'$ at points $E$ and $F$, respectively, and point $G$ is the intersection of lines $AF$ and $BH$. Prove that $CE=DT$ where $T$ is the intersection of $GE$ and $AD$.

Let ABC be an isosceles triangle with $AB = AC$ and altitude $BD$. If $CD: AD = 1: 2$ , prove that $BC^2 = \frac23 AC^2$

Let $ABC$ be a triangle with points $D, E$, and $F$ on the line segments $AB, AC$, and $BC$ respectively such that $BFED$ is a parallelogram. Points $H,G$ lie n the extensions of $DE$ ,$DF$ beyond $E, F$ respectively such that $HE: GE: FE = 1: 5: 8$, $HG = GF$ and $\angle BDF = \angle DHG$. Find the ratio  $HE: HG$.

Let $ABCD$ be an inscribed quadrilateral , such that the diameter of the circumcircle $AC$ has length $10$ units. The diagonals $AC$ and $BD$ intersects at point $M$ and the length of $AM$ is $4$ units. Let the line $XY$ be the tangent line of the circle at point $A$. Extensions of sides $CD$ and $CB$ intersect the line $XY$ at the points $P$ and $Q$ respectively. Find the value of $AP \cdot  AQ$.

Given $\Delta ABC$ and its centroid $G$, If line $AC$ is tangent to $\odot (ABG)$. Prove that,\begin{align*} AB+BC \leq 2AC \end{align*}

Given triangle $ABC$ , points $D$ and $F$ lie on sides $BC$ and $AB$ respectively, such that $BD = 7$, $DC = 2$, $BF = 5$, $FA = 2$ and $AD$ intersects $FC$ at point $P$. If $PC = 3$, then find the lengths of $AP$ and $PD$.

A quadrilateral $ABCD$ inscribed in a circle. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$ . If the diagonal $AC$, the diagonal $BD$ and the line $MN$ intersect at one point, then angle $BAD$ is equal to angle $ABC$.

Gives a triangle $ABC$ inscribed in a circle of radius $5$ units. Let O be it's orthocenter. If the side $BC$ is $8$ units, then find the length of $AO$.

Given a $\Delta ABC$ where $\angle C = 90^{\circ}$, $D$ is a point in the interior of $\Delta ABC$ and lines $AD$ $,$ $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ at points $P$ ,$Q$ and $R$ ,respectively. Let $M$ be the midpoint of $\overline{PQ}$. Prove that, if $\angle BRP$ $ =$ $ \angle PRC$ then $MR=MC$.

Let $I$ be the center of the inscribed circle of triangle $ABC$ and $AI, BI, CI$ intersect the sides $BC, CA, AB$ at points $A_1, B_1, C_1$, respectively. Prove that$$\frac{AI}{A_1I} \frac{BI}{B_1I} \frac{CI}{C_1I}\ge 8.$$

Let $ABC$ be a triangle where angles $ABC$ and $BAC$ are acute. The bisector of internal and external angles of angle $BAC$ intersect the line $BC$ at points $D$ and $E$, respectively. Let $O$ be the center of the circumcscribed circle of the triangle $ADE$. If point $P$ lie on this circle of center $O$, prove that$$\frac{BP}{PC}=\frac{OB}{OA}$$

Circles of radii $r_1, r_2$ and $r_3$ are externally touching each other at points $A, B$, and $C$. If the triangle $ABC$ has perimeter equal to $p$, prove that$$\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\ge \frac{9}{p}.$$

In $\Delta ABC$, Let the Incircle touch $\overline{BC}, \overline{CA}, \overline{AB}$ at $X,Y,Z$. Let $I_A,I_B,I_C$ be $A$,$B$,$C-$excenters, respectively. Prove that Incenter of $\Delta ABC$, orthocenter of $\Delta XYZ$ and centroid of $\Delta I_AI_BI_C$ are collinear.

1Let $ABC$ be a right triangle with $\angle BAC$ right angle and $AB = \frac12 BC$. Let $D$ be the midpoint of $BC$ and $E$ be the point on the same semiplane with $A$ wrt the line $BC$ such that $DE = AB$. From point $E$ the perpendicular on $AC$ cuts it at point $F$. Line $DF$ intersects $AE$ at point $G$. Prove that $GD$ is perpendicular to $AE$

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$. Let $P$ be a point on side $BC$, and let $\omega$ be the circle with diameter $CP$. Suppose that $\omega$ intersects $AC $and $AP$ again at $Q$ and $R$, respectively. Show that $CP^2 = AC \cdot CQ - AP \cdot P R$.

Let $ABC$ be a triangle with $\angle ABC = 45^o$. Point $P$ lies on the side BC that $PC=2$ units and $\angle BAP = 15^o$. If the line segment of the tangent drawn from point $B$ to the circumcircle of the triangle $APC$ is $\sqrt3$ units, find the measure of the $\angle ACP$.

Let $ABCD$ be an cyclic quadrilateral. Let the diagonal $AC$ and $BD$ meet at point $X$. Let $Z$ be point on $AD$ such that $AZ = ZX$. The line $ZX$ intersects the side $BC$ at the point $Y$. If $AD^2 = 2DX^2$ , prove that $BY = YC$.

Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.

Let $A_1A_2A_3$ be a triangle with incenter $I$. Let the inner angle bisectors of $ \angle A_1, \angle A_2, \angle A_3$ intersect the circumcircle again at $B_1, B_2, B_3$ respectively. Prove that$$\frac{IA_i^2}{IB_i}\ge 2r$$for any $i\in \{1,2,3\}$ , where $r$ is the radius of the inscribed circle of the triangle $A_1A_2A_3$ .

Let $ABC$ be a triangle, with $I$ the center of the inscribed circle, with touches sides $AB$ and $BC$ at points $D$ and $E$ respectively. Let $K$ and $L$ be points on the incircle, such that $DK$ and $EL$ are diameters of the incircle. If $AB + BC = 3AC$, prove that $A,C,I,K,L$ lie on the same circle.
Let $ABC$ be a triangle with $AC> BC$. A circle that passes through point $A$ and touches the side $BC$ at point $B$, intersects line $AC$ at $D\ne A$. Line $BD$ intersects the circumcircle of triangle $ABC$ again at point $E$. Let $F$ be the point on the circumscribed circle of triangle $CDE$ such that $DF = DB$. Prove that $F$ lies on the line $AE$ or $BC$.

Let $\ell$ be the common tangent of $\omega_1$ and $\omega_2$ which is tangent at $\omega_1$ and $\omega_2$ at points $A$ and $B$ respectively where the circles $\omega_1$ and $\omega_2$ lie on the same side wrt $\ell$ . Let $M$ be the midpoint of $AB$. From point $M$ draw a tangent to $\omega_1$ that intersects it at point $C\ne A$. From point $M$ draw a tangent to $\omega_2$ that intersects it at point $D\ne B$. Let $P$ be a point on $O_1O_2$ such that $MP \perp O_1O_2 $. Show that the circumscribed circle of the triangle $CPD$ is tangent to the circles $\omega_1$ and $\omega_2$.

Let $\vartriangle ABC$ be an acute triangle, and let $P$ be the foot of altitude from $C$ to $AB$. Let $\omega$ be the circle with diameter $BC$. The tangents from $A$ to $\omega$ are drawn touching $\omega$ at $D$ and $E$. Lines $AD$ and $AE$ intersect line $BC$ at $M$ and $N$ respectively, so that $B$ lies between $M$ and $C$. Let $CP$ intersect $DE$ at $Q, ME$ intersect $ND$ at $R$, and let $QR$ intersect $BC$ at $S$. Show that $QS$ bisects $\angle DSE$

Let the line segment $AB$ be a common chord of two circles with center $O_1$ and $O_2$ ($O_1 \ne O_2$). Let $k_1$ and $k_2$ represent the arc $AB$ on the same side wrt line $AB$ of the circle $O_1$ and $O_2$ respectively, with $k_1$ between $k_2$ and $AB$. Let $X$ be the point on $k_2$ that does not lie on the perpendicular bisector of the segment $AB$. Tangent at $X$ of $k_2$ intersects $AB$ at point $C$. Let $Y$ be a point on $k_1$ such that $CX = CY$. Show that the line $XY$ passes through a fixed point independent of the position of $X$ on $k_2$.

Let $ABCD$ be a square with side $1$, with $P$ and $Q$ being points on the sides $AB$ and $BC$, respectively, such that $PB + BQ = 1$. If $PC$ intersects $AQ$ at $E$, prove that the line $DE$ is perpendicular to the line $PQ$.

$ABC$ is an acute triangle with $D, E$ and $F$ being the feet of the altitudes of the triangle $ABC$ on sides $BC, AC$ and $AB$ respectively. Let $P, Q$ and $R$ be the midpoints of $DE, EF$ and $FD$ respectively. Then show that the lines passing through $P, Q$, and $R$ perpendicular on sides $AB, BC$, and $CA$, respectively, intersect at a single point.

Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot  DE = BD \cdot CE$

Let $ABC$ be an isosceles triangle with $A$ being the apex, less than $60^o$ with $D$ the point on the side $AC$ , such that $\angle DBC =  \angle BAC$. Let $L_1$ be a line passing through point $A$ and parallel to side $BC$. Let $L_2$ be the perpendicular bisector of side $BD$. $L_1$ and $L_2$ intersect at point $E$. show that the $EC$ is bisected by $AB$.

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let the tangents of circle $\Gamma$ at points $B$ and $C$ intersect at point $D$ . Let $M$ be the point on the side $BC$ such that $\angle BAM = \angle CAD$. Prove that the center of circle $\Gamma$ lies on the line $MD$.

Let A$BC$ be a triangle with $A$ right angle and $D$ is a point on the side $BC$ such that $AD$ is perpendicular to the side $BC$. Let $W_1$ and $W_2$ are the centers of the incircles of the triangles $ABD$ and $ADC$ respectively. Line $W_1W_2$ intersects $AB$ at $X$ and $AC$ at $Y$. Prove that $AX = AD = AY$.

Let $ABC$ be an acute triangle with $E$ and $F$ on sides $AB$ and $AC$ respectively, and $O$ be it's circumcenter. Let $AO$ intersect $BC$ at point $D$. Let the perpendicular from point $D$ on sides $AB$ and $AC$ intersect them at points $M,N$ respectively. Let the perpendicular on the side $BC$ from points $E,F,M,N$ intersect it at points $E',F',M',N'$ respectively. Prove that $A, D, E, F$ lie on the same circle if and only if $E'F '= M'N'$.

Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:
For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle  BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.


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