geometry problems from 3rd Round of Iranian Mathematical Olympiads
with aops links in the names
Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.
2017 Iran MO 3rd Round first geometry P2
Let $ABCD$ be a trapezoid ($AB<CD,AB\parallel CD$) and $P\equiv AD\cap BC$. Suppose that $Q$ be a point inside $ABCD$ such that $\angle QAB=\angle QDC=90-\angle BQC$. Prove that $\angle PQA=2\angle QCD$.
with aops links in the names
1996 - 2020
(missing 1999, 2001)
Let $ABCD$ be a parallelogram. Construct the equilateral triangle $DCE$ on the side $DC$ and outside of parallelogram. Let $P$ be an arbitrary point in plane of $ABCD$. Show that $ PA+PB+AD \geq PE.$
Let $ABC$ and $XYZ$ be two triangles. Define $A_1=BC\cap ZX, A_2=BC\cap XY, B_1=CA\cap XY, B_2=CA\cap YZ, C_1=AB\cap YZ, C_2=AB\cap ZX.$ Hereby, the abbreviation $g\cap h$ means the point of intersection of two lines $g$ and $h$.
Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.
Let $ABCD$ be a convex pentagon such that $\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.$ Let $F$ be a point on AB such that $AF:BF=AE:BC$. Show that $\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.$
$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.
In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB=CD$, $ AC=BD$ and $ AD= BC$.
Suppose that $ M$ is an arbitrary point on side $ BC$ of triangle $ ABC$. $ B_1,C_1$ are points on $ AB,AC$ such that $ MB = MB_1$ and $ MC = MC_1$. Suppose that $ H,I$ are orthocenter of triangle $ ABC$ and incenter of triangle $ MB_1C_1$. Prove that $ A,B_1,H,I,C_1$ lie on a circle.
(missing 1999, 2001)
Let $ABCD$ be a convex quadrilateral. Construct the points $P,Q,R,$ and $S$ on continue of $AB,BC,CD,$ and $DA$, respectively, such that $BP=CQ=DR=AS.$ Show that if $PQRS$ is a square, then $ABCD$ is also a square.
Consider a semicircle of center $O$ and diameter $AB$. A line intersects $AB$ at $M$ and the semicircle at $C$ and $D$ s.t. $MC>MD$ and $MB<MA$. The circumcircles od the $AOC$ and $BOD$ intersect again at $K$. Prove that $MK\perp KO$.
Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.
In an acute triangle $ABC$, points $D,E,F$ are the feet of the altitudes from $A,B,C$, respectively. A line through $D$ parallel to $EF$ meets $AC$ at $Q$ and $AB$ at $R$. Lines $BC$ and $EF$ intersect at $P$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of $BC$.
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE$ and $EF = FA$. Prove that
$\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.$
$\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.$
In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$
1999 problems are missing from aops
Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.
Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN = AE + AF$.
Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN = AE + AF$.
2000 Iran MO 3rd Round geometry 3 of 6
Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 + a'^2}{3}}$.
Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.
Isosceles triangles $A_3A_1O_2$ and $A_1A_2O_3$ are constructed on the sides of a triangle $A_1A_2A_3$ as the bases, outside the triangle. Let $O_1$ be a point outside $\Delta A_1A_2A_3$ such that $\angle O_1A_3A_2 =\frac 12\angle A_1O_3A_2$ and $\angle O_1A_2A_3 =\frac 12\angle A_1O_2A_3$. Prove that $A_1O_1\perp O_2O_3$, and if $T$ is the projection of $O_1$ onto $A_2A_3$, then $\frac{A_1O_1}{O_2O_3} = 2\frac{O_1T}{A_2A_3}$.
A circle$\Gamma$ with radius $R$ and center $\omega$, and a line $d$ are drawn on a plane, such that the distance of $\omega$ from $d$ is greater than $R$. Two points $M$ and $N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\Gamma$. Prove that there is a point $P$ in the plane from which all the segments $MN$ are visible at a constant angle.
2001 problems are missing from aops
$M$ is midpoint of $BC$.$P$ is an arbitary point on $BC$. $C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$. Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P. $C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$. Prove : 4r_{1}+r_{2}+r_{3}=R$ ($R$ radius of big circle)
In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?
Circles $C_{1}$ and $C_{2}$ are tangent to each other at $K$ and are tangent to circle $C$ at $M$ and $N$. External tangent of $C_{1}$ and $C_{2}$ intersect $C$ at $A$ and $B$. $AK$ and $BK$ intersect with circle $C$ at $E$ and $F$ respectively. If AB is diameter of $C$, prove that $EF$ and $MN$ and $OK$ are concurrent. ($O$ is center of circle $C$.)
Let $ M$ and $ N$ be points on the side $ BC$ of triangle $ ABC$, with the point $ M$ lying on the segment $ BN$, such that $ BM =CN$. Let $ P$ and $ Q$ be points on the segments $ AN$ and $ AM$, respectively, such that $ \angle PMC = \angle MAB$ and $ \angle QNB =\angle NAC$. Prove that $ \angle QBC = \angle PCB$.
$H,I,O,N$ are orthogonal center, incenter, circumcenter, and Nagelian point of triangle $ABC$. $I_{a},I_{b},I_{c}$ are excenters of $ABC$ corresponding vertices $A,B,C$. $S$ is point that $O$ is midpoint of $HS$. Prove that centroid of triangles $I_{a}I_{b}I_{c}$ and $SIN$ concide.
Let $A$ be be a point outside the circle $C$, and $AB$ and $AC$ be the two tangents from $A$ to this circle $C$. Let $L$ be an arbitrary tangent to $C$ that cuts $AB$ and $AC$ in $P$ and $Q$. A line through $P$ parallel to $AC$ cuts $BC$ in R. Prove that while $L$ varies, $QR$ passes through a fixed point.
$I$ is incenter of triangle $ABC$. Incircle of $ABC$ touches $AB,AC$ at $X,Y$. $XI$ intersects incircle at $M$. Let $CM\cap AB=X'$. $L$ is a point on the segment $X'C$ that $X'L=CM$. Prove that $A,L,I$ are collinear iff $AB=AC$.
Excircle of triangle $ABC$ corresponding vertex $A$, is tangent to $BC$ at $P$. $AP$ intersects circumcircle of $ABC$ at $D$. Prove $r(PCD)=r(PBD)$, where $r(PCD)$ and $r(PBD)$ are inradii of triangles $PCD$ and $PBD$.
2002 Iran MO 3rd Round geometry 10 of 10
$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) . Prove that $P$ and $I$ and $D$ are on a line.
Let $ ABC$ be a triangle. $ W_a$ is a circle with center on $ BC$ passing through $ A$ and perpendicular to circumcircle of $ ABC$. $ W_b,W_c$ are defined similarly. Prove that center of $ W_a,W_b,W_c$ are collinear.
$ A,B$ are fixed points. Variable line $ l$ passes through the fixed point $ C$. There are two circles passing through $ A,B$ and tangent to $ l$ at $ M,N$. Prove that circumcircle of $ AMN$ passes through a fixed point.
Let $ A,B,C,Q$ be fixed points on plane. $ M,N,P$ are intersection points of $ AQ,BQ,CQ$ with $ BC,CA,AB$. $ D',E',F'$ are tangency points of incircle of $ ABC$ with $ BC,CA,AB$. Tangents drawn from $ M,N,P$ (not triangle sides) to incircle of $ ABC$ make triangle $ DEF$. Prove that $ DD',EE',FF'$ intersect at $ Q$.
Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.
Assume $ABCD$ a convex quadrilatral. $P$ and $Q$ are on $BC$ and $DC$ respectively such that $\angle BAP= \angle DAQ$ .prove that $[ADQ]=[ABP] $ ($[ABC]$ means its area ) iff the line which crosses through the orthocenters of these triangles , is perpendicular to $AC$.
$\angle XOY$ is angle in the plane. $A,B$ are variable points on $OX,OY$ such that $1/OA+1/OB=1/k$ ($k$ is constant). Draw two circles with diameters $OA$ and $OB$. Prove that common external tangent to these circles is tangent to a constant circle. (Determine the radius and the locus of its center)..
Let the incircle of a triangle $ABC$ touch $BC,AC,AB$ at $A_1,B_1,C_1$ respectively. $M$ and $N$ are the midpoints of $AB_1$ and $AC_1$ respectively. $MN$ meets $A_1C_1$ at $T$ . draw two tangents $TP$ and $TQ$ through $T$ to incircle. $PQ$ meets $MN$ at $L$ and $B_1C_1$ meets $PQ$ at $K$ . Assume $I$ is the center of the incircle . Prove $IK$ is parallel to $AL$
Let $ABC$ be a triangle, and $O$ the center of its circumcircle. Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively. Prove that $\angle ROS=\angle BAC$.
Assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let $AP_a$ be a diameter of the circumcircle. the pependicular from A' to $AP_a$ meets the tangent to circumcircle at A" in the point $X_a$; we define $X_b,X_c$ similary . prove that $X_a,X_b,X_c$ are one a line.
Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .
In triangle $ ABC$, points $ M,N$ lie on line $ AC$ such that $ MA=AB$ and $ NB= NC$. Also $ K,L$ lie on line $ BC$ such that $ KA=KB$ and $ LA=LC$. It is know that $ KL= \frac12{BC}$ and $ MN=AC$. Find angles of triangle $ ABC$. [might have a typo]
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB+AC=3BC$.
From each vertex of triangle $ABC$ we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them $\ell_1$, $\ell_2$ and $\ell_3$.
a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$.
b) Find the locus of $P$ as we move the 3 arbitary lines.
a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$.
b) Find the locus of $P$ as we move the 3 arbitary lines.
Suppose $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is $ \frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}$
Prove that in acute-angled traingle ABC if $r$ is inradius and $R$ is radius of circumcircle then: $a^2+b^2+c^2\geq 4(R+r)^2$
Suppose in triangle $ABC$ incircle touches the side $BC$ at $P$ and $\angle APB=\alpha$. Prove that : $\frac1{p-b}-\frac1{p-c}=\frac2{rtg\alpha}$
Suppose $H$ and $O$ are orthocenter and circumcenter of triangle $ABC$. $\omega$ is circumcircle of $ABC$. $AO$ intersects with $\omega$ at $A_1$. $A_1H$ intersects with $\omega$ at $A'$ and $A''$ is the intersection point of $\omega$ and $AH$. We define points $B',\ B'',\ C'$ and $C''$ similiarly. Prove that $A'A'',B'B''$ and $C'C''$ are concurrent in a point on the Euler line of triangle $ABC$.
Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.
$ABC$ is a triangle and $R,Q,P$ are midpoints of $AB,AC,BC$. Line $AP$ intersects $RQ$ in $E$ and circumcircle of $ABC$ in $F$. $T,S$ are on $RP,PQ$ such that $ES\perp PQ,ET\perp RP$. $F'$ is on circumcircle of $ABC$ that $FF'$ is diameter. The point of intersection of $AF'$ and $BC$ is $E'$. $S',T'$ are on $AB,AC$ that $E'S'\perp AB,E'T'\perp AC$. Prove that $TS$ and $T'S'$ are perpendicular.
In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that $LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})$
Circle $\Omega(O,R)$ and its chord $AB$ is given. Suppose $C$ is midpoint of arc $AB$. $X$ is an arbitrary point on the cirlce. Perpendicular from $B$ to $CX$ intersects circle again in $D$. Perpendicular from $C$ to $DX$ intersects circle again in $E$. We draw three lines $\ell_{1},\ell_{2},\ell_{3}$ from $A,B,E$ parralell to $OX,OD,OC$. Prove that these lines are concurrent and find locus of concurrncy point.
$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that $\cos B+\cos C=1\Longleftrightarrow MI=MT$
a) Let $ ABC$ be a triangle, and $ O$ be its circumcenter. $ BO$ and $ CO$ intersect with $ AC,AB$ at $ B',C'$. $ B'C'$ intersects the circumcircle at two points $ P,Q$. Prove that $ AP= AQ$ if and only if $ ABC$ is isosceles.
b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.
a) Let $ ABC$ be a triangle, and $ O$ be its circumcenter. $ BO$ and $ CO$ intersect with $ AC,AB$ at $ B',C'$. $ B'C'$ intersects the circumcircle at two points $ P,Q$. Prove that $ AP =AQ$ if and only if $ ABC$ is isosceles.
b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.
b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.
b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.
Let $ I$ be incenter of triangle $ ABC$, $ M$ be midpoint of side $ BC$, and $ T$ be the intersection point of $ IM$ with incircle, in such a way that $ I$ is between $ M$ and $ T$. Prove that $ \angle BIM -\angle CIM= \frac{3}2(\angle B -\angle C)$, if and only if $ AT\perp BC$.
Let $ ABC$ be a triangle, and $ D$ be a point where incircle touches side $ BC$. $ M$ is midpoint of $ BC$, and $ K$ is a point on $ BC$ such that $ AK\perp BC$. Let $ D'$ be a point on $ BC$ such that $ \frac{D'M}{D'K}=\frac{DM}{DK}$. Define $ \omega_{a}$ to be circle with diameter $ DD'$. We define $ \omega_{B},\omega_{C}$ similarly. Prove that every two of these circles are tangent.
Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.
Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' = BB' = CC' = x$. Prove that:
a) If $ ABC\sim A'B'C'$ then $ x = 2r$
b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x =R + d$ or $ x = R- d$.
(In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d = OI$)
Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.
a) If $ ABC\sim A'B'C'$ then $ x = 2r$
b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x =R + d$ or $ x = R- d$.
(In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d = OI$)
Let $ ABCD$ be a quadrilateral, and $ E$ be intersection points of $ AB,CD$ and $ AD,BC$ respectively. External bisectors of $ DAB$ and $ DCB$ intersect at $ P$, external bisectors of $ ABC$ and $ ADC$ intersect at $ Q$ and external bisectors of $ AED$ and $ AFB$ intersect at $ R$. Prove that $ P,Q,R$ are collinear.
Let $ ABC$ be an isosceles triangle with $ AB=AC$, and $ D$ be midpoint of $ BC$, and $ E$ be foot of altitude from $ C$. Let $ H$ be orthocenter of $ ABC$ and $ N$ be midpoint of $ CE$. $ AN$ intersects with circumcircle of triangle $ ABC$ at $ K$. The tangent from $ C$ to circumcircle of $ ABC$ intersects with $ AD$ at $ F$. Suppose that radical axis of circumcircles of $ CHA$ and $ CKF$ is $ BC$. Find $ \angle BAC$.
Let $ ABC$ be an isosceles triangle with $ AB=AC$, and $ D$ be midpoint of $ BC$, and $ E$ be foot of altitude from $ C$. Let $ H$ be orthocenter of $ ABC$ and $ N$ be midpoint of $ CE$. $ AN$ intersects with circumcircle of triangle $ ABC$ at $ K$. The tangent from $ C$ to circumcircle of $ ABC$ intersects with $ AD$ at $ F$. Suppose that radical axis of circumcircles of $ CHA$ and $ CKF$ is $ BC$. Find $ \angle BAC$.
Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP = AT$.$ P$ and $ T$ are at the same side wrt $ A$. Prove $ \angle APD= 90^\circ$.
2009 Iran MO 3rd Round geometry P2
There is given a trapezoid $ ABCD$.We have the following properties:$ AD\parallel{}BC,DA =DB= DC,\angle BCD = 72^\circ$. A point $ K$ is taken on $ BD$ such that $ AD = AK,K \neq D$.Let $ M$ be the midpoint of $ CD$.$ AM$ intersects $ BD$ at $ N$.PROVE $ BK= ND$.
There is given a trapezoid $ ABCD$.We have the following properties:$ AD\parallel{}BC,DA =DB= DC,\angle BCD = 72^\circ$. A point $ K$ is taken on $ BD$ such that $ AD = AK,K \neq D$.Let $ M$ be the midpoint of $ CD$.$ AM$ intersects $ BD$ at $ N$.PROVE $ BK= ND$.
There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD = 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.
Point $ P$ is taken on the segment $ BC$ of the scalene triangle $ ABC$ such that $ AP \neq AB,AP \neq AC$.$ l_1,l_2$ are the incenters of triangles $ ABP,ACP$ respectively. circles $ W_1,W_2$ are drawn centered at $ l_1,l_2$ and with radius equal to $ l_1P,l_2P$,respectively. $ W_1,W_2$ intersects at $ P$ and $ Q$. $ W_1$ intersects $ AB$ and $ BC$ at $ Y_1$ ( the intersection closer to $B$) and $ X_1$,respectively. $ W_2$ intersects $ AC$ and $ BC$ at $ Y_2$ (the intersection closer to $C$) and $ X_2$,respectively. Prove the concurrency of $ PQ,X_1Y_1,X_2Y_2$.
Two circles $ S_1$ and $ S_2$ with equal radius and intersecting at two points are given in the plane.A line $ l$ intersects $ S_1$ at $ B,D$ and $ S_2$ at $ A,C$(the order of the points on the line are as follows:$ A,B,C,D$).Two circles $ W_1$ and $ W_2$ are drawn such that both of them are tangent externally at $ S_1$ and internally at $ S_2$ and also tangent to $ l$ at both sides.Suppose $ W_1$ and $ W_2$ are tangent.Then PROVE $ AB =CD$.
In a triangle $ABC$, $O$ is the circumcenter and $I$ is the incenter. $X$ is the reflection of $I$ to $O$. $A_1$ is foot of the perpendicular from $X$ to $BC$. $B_1$ and $C_1$ are defined similarly. prove that $AA_1$,$BB_1$ and $CC_1$ are concurrent.
In a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$.
a) in what ratio does $EF$ cut the digonals?
b) find $\frac{AF}{FD}$.
a) in what ratio does $EF$ cut the digonals?
b) find $\frac{AF}{FD}$.
in a quadrilateral $ABCD$ digonals are perpendicular to each other. let $S$ be the intersection of digonals. $K$,$L$,$M$ and $N$ are reflections of $S$ to $AB$,$BC$,$CD$ and $DA$. $BN$ cuts the circumcircle of $SKN$ in $E$ and $BM$ cuts the circumcircle of $SLM$ in $F$. prove that $EFLK$ is concyclic.
in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.
In a triangle $ABC$, $I$ is the incenter. $D$ is the reflection of $A$ to $I$. the incircle is tangent to $BC$ at point $E$. $DE$ cuts $IG$ at $P$ ($G$ is centroid). $M$ is the midpoint of $BC$. prove that
a) $AP||DM$.
b) $AP=2DM$.
a) $AP||DM$.
b) $AP=2DM$.
In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.
We have $4$ circles in plane such that any two of them are tangent to each other. we connect the tangency point of two circles to the tangency point of two other circles. Prove that these three lines are concurrent.
by Masoud Nourbakhsh
2011 Iran MO 3rd Round geometry P2
In triangle $ABC$, $\omega$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $\omega$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Prove that $PQ$ is tangent to $\omega$ iff $OM=ON$.
2012 Iran MO 3rd Round geometry P1In triangle $ABC$, $\omega$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $\omega$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Prove that $PQ$ is tangent to $\omega$ iff $OM=ON$.
by Mr.Etesami
2011 Iran MO 3rd Round geometry P3
In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear.
In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear.
by Amirhossein Zabeti
2011 Iran MO 3rd Round geometry P4
A variant triangle has fixed incircle and circumcircle. Prove that the radical center of its three excircles lies on a fixed circle and the circle's center is the midpoint of the line joining circumcenter and incenter.
A variant triangle has fixed incircle and circumcircle. Prove that the radical center of its three excircles lies on a fixed circle and the circle's center is the midpoint of the line joining circumcenter and incenter.
by Masoud Nourbakhsh
2011 Iran MO 3rd Round geometry P5
Given triangle $ABC$, $D$ is the foot of the external angle bisector of $A$, $I$ its incenter and $I_a$ its $A$-excenter. Perpendicular from $I$ to $DI_a$ intersects the circumcircle of triangle in $A'$. Define $B'$ and $C'$ similarly. Prove that $AA',BB'$ and $CC'$ are concurrent.
by Amirhossein Zabeti
Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant.
by Mehdi E'tesami Fard
Let the Nagel point of triangle $ABC$ be $N$. We draw lines from $B$ and $C$ to $N$ so that these lines intersect sides $AC$ and $AB$ in $D$ and $E$ respectively. $M$ and $T$ are midpoints of segments $BE$ and $CD$ respectively. $P$ is the second intersection point of circumcircles of triangles $BEN$ and $CDN$. $l_1$ and $l_2$ are perpendicular lines to $PM$ and $PT$ in points $M$ and $T$ respectively. Prove that lines $l_1$ and $l_2$ intersect on the circumcircle of triangle $ABC$.
by Nima Hamidi
Consider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that $\frac{TP}{TQ}\ge \frac{b}{a}.$
by Morteza Saghafian
The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$.
by Pedram Safaei
Two fixed lines $l_1$ and $l_2$ are perpendicular to each other at a point $Y$. Points $X$ and $O$ are on $l_2$ and both are on one side of line $l_1$. We draw the circle $\omega$ with center $O$ and radius $OY$. A variable point $Z$ is on line $l_1$. Line $OZ$ cuts circle $\omega$ in $P$. Parallel to $XP$ from $O$ intersects $XZ$ in $S$. Find the locus of the point $S$.
by Nima Hamidi
2013 Iran MO 3rd Round geometry P1
Let $ABCDE$ be a pentagon inscribe in a circle $(O)$. Let $ BE \cap AD = T$. Suppose the parallel line with $CD$ which passes through $T$ which cut $AB,CE$ at $X,Y$. If $\omega$ be the circumcircle of triangle $AXY$ then prove that $\omega$ is tangent to $(O)$.
Let $ABCDE$ be a pentagon inscribe in a circle $(O)$. Let $ BE \cap AD = T$. Suppose the parallel line with $CD$ which passes through $T$ which cut $AB,CE$ at $X,Y$. If $\omega$ be the circumcircle of triangle $AXY$ then prove that $\omega$ is tangent to $(O)$.
2013 Iran MO 3rd Round geometry P2
Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.
Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.
2013 Iran MO 3rd Round geometry P3
Suppose line $\ell$ and four points $A,B,C,D$ lies on $\ell$. Suppose that circles $\omega_1 , \omega_2$ passes through $A,B$ and circles $\omega'_1 , \omega'_2$ passes through $C,D$. If $\omega_1 \perp \omega'_1$ and $\omega_2 \perp \omega'_2$ then prove that lines $O_1O'_2 , O_2O'_1 , \ell $ are concurrent where $O_1,O_2,O'_1,O'_2$ are center of $\omega_1 , \omega_2 , \omega'_1 , \omega'_2$
Suppose line $\ell$ and four points $A,B,C,D$ lies on $\ell$. Suppose that circles $\omega_1 , \omega_2$ passes through $A,B$ and circles $\omega'_1 , \omega'_2$ passes through $C,D$. If $\omega_1 \perp \omega'_1$ and $\omega_2 \perp \omega'_2$ then prove that lines $O_1O'_2 , O_2O'_1 , \ell $ are concurrent where $O_1,O_2,O'_1,O'_2$ are center of $\omega_1 , \omega_2 , \omega'_1 , \omega'_2$
2013 Iran MO 3rd Round geometry P4
In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.
In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.
2013 Iran MO 3rd Round geometry P5
Let $ABC$ be triangle with circumcircle $(O)$. Let $AO$ cut $(O)$ again at $A'$. Perpendicular bisector of $OA'$ cut $BC$ at $P_A$. $P_B,P_C$ define similarly. Prove that :
i) Point $P_A,P_B,P_C$ are collinear.
ii ) Prove that the distance of $O$ from this line is equal to $\frac {R}{2}$ where $R$ is the radius of the circumcircle.
Let $ABC$ be triangle with circumcircle $(O)$. Let $AO$ cut $(O)$ again at $A'$. Perpendicular bisector of $OA'$ cut $BC$ at $P_A$. $P_B,P_C$ define similarly. Prove that :
i) Point $P_A,P_B,P_C$ are collinear.
ii ) Prove that the distance of $O$ from this line is equal to $\frac {R}{2}$ where $R$ is the radius of the circumcircle.
2014 Iran MO 3rd Round geometry P1
In the circumcircle of triange $\triangle ABC,$ $AA'$ is a diameter. We draw lines $l'$ and $l$ from $A'$ parallel with Internal and external bisector of the vertex $A$. $l'$ Cut out $AB , BC$ at $B_1$ and $B_2$. $l$ Cut out $AC , BC$ at $C_1$ and $C_2$. Prove that the circumcircles of $\triangle ABC$ $\triangle CC_1C_2$ and $\triangle BB_1B_2$ have a common point.
In the circumcircle of triange $\triangle ABC,$ $AA'$ is a diameter. We draw lines $l'$ and $l$ from $A'$ parallel with Internal and external bisector of the vertex $A$. $l'$ Cut out $AB , BC$ at $B_1$ and $B_2$. $l$ Cut out $AC , BC$ at $C_1$ and $C_2$. Prove that the circumcircles of $\triangle ABC$ $\triangle CC_1C_2$ and $\triangle BB_1B_2$ have a common point.
$\triangle{ABC}$ is isosceles$(AB=AC)$. Points $P$ and $Q$ exist inside the triangle such that $Q$ lies inside $\widehat{PAC}$ and $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$. We also have $BP=PQ=CQ$.Let $X$ and $Y$ be the intersection points of $(AP,BQ)$ and $(AQ,CP)$ respectively. Prove that quadrilateral $PQYX$ is cyclic.
Distinct points $B,B',C,C'$ lie on an arbitrary line $\ell$. $A$ is a point not lying on $\ell$. A line passing through $B$ and parallel to $AB'$ intersects with $AC$ in $E$ and a line passing through $C$ and parallel to $AC'$ intersects with $AB$ in $F$. Let $X$ be the intersection point of the circumcircles of $\triangle{ABC}$ and $\triangle{AB'C'}$($A \neq X$). Prove that $EF \parallel AX$.
$D$ is an arbitrary point lying on side $BC$ of $\triangle{ABC}$. Circle $\omega_1$ is tangent to segments $AD$ , $BD$ and the circumcircle of $\triangle{ABC}$ and circle $\omega_2$ is tangent to segments $AD$ , $CD$ and the circumcircle of $\triangle{ABC}$. Let $X$ and $Y$ be the intersection points of $\omega_1$ and $\omega_2$ with $BC$ respectively and take $M$ as the midpoint of $XY$. Let $T$ be the midpoint of arc $BC$ which does not contain $A$. If $I$ is the incenter of $\triangle{ABC}$, prove that $TM$ goes through the midpoint of $ID$.
$X$ and $Y$ are two points lying on or on the extensions of side $BC$ of $\triangle{ABC}$ such that $\widehat{XAY} = 90$. Let $H$ be the orthocenter of $\triangle{ABC}$. Take $X'$ and $Y'$ as the intersection points of $(BH,AX)$ and $(CH,AY)$ respectively. Prove that circumcircle of $\triangle{CYY'}$,circumcircle of $\triangle{BXX'}$ and $X'Y'$ are concurrent.
2015 Iran MO 3rd Round geometry P1
Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.
Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.
2015 Iran MO 3rd Round geometry P2
Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.
Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.
2015 Iran MO 3rd Round geometry P3
Let $ABC$ be a triangle. consider an arbitrary point $P$ on the plain of $\triangle ABC$. Let $R,Q$ be the reflections of $P$ wrt $AB,AC$ respectively. Let $RQ\cap BC=T$. Prove that $\angle APB=\angle APC$ if and if only $\angle APT=90^{\circ}$.
Let $ABC$ be a triangle. consider an arbitrary point $P$ on the plain of $\triangle ABC$. Let $R,Q$ be the reflections of $P$ wrt $AB,AC$ respectively. Let $RQ\cap BC=T$. Prove that $\angle APB=\angle APC$ if and if only $\angle APT=90^{\circ}$.
2015 Iran MO 3rd Round geometry P4
Let $ABC$ be a triangle with incenter $I$. Let $K$ be the midpoint of $AI$ and $BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N$. points $P,Q$ lie on $AM,AN$ respectively such that $\angle ABK=\angle PBC,\angle ACK=\angle QCB$. Prove that $P,Q,I$ are collinear.
Let $ABC$ be a triangle with incenter $I$. Let $K$ be the midpoint of $AI$ and $BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N$. points $P,Q$ lie on $AM,AN$ respectively such that $\angle ABK=\angle PBC,\angle ACK=\angle QCB$. Prove that $P,Q,I$ are collinear.
Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $R$ be the radius of circumcircle of $\triangle ABC$. Let $A',B',C'$ be the points on $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ respectively such that $AH.AA'=R^2,BH.BB'=R^2,CH.CC'=R^2$. Prove that $O$ is incenter of $\triangle A'B'C'$.
Let $ABC$ be an arbitrary triangle,$P$ is the intersection point of the altitude from $C$ and the tangent line from $A$ to the circumcircle. The bisector of angle $A$ intersects $BC$ at $D$ . $PD$ intersects $AB$ at $K$, if $H$ is the orthocenter then prove : $HK\perp AD$
Let $ABC$ be an arbitrary triangle. Let $E,E$ be two points on $AB,AC$ respectively such that their distance to the midpoint of $BC$ is equal. Let $P$ be the second intersection of the triangles $ABC,AEF$ circumcircles . The tangents from $E,F$ to the circumcircle of $AEF$ intersect each other at $K$. Prove that : $\angle KPA = 90$
Let $ABC$ be a triangle and let $AD,BE,CF$ be its altitudes . $FA_{1},DB_{1},EC_{1}$ are perpendicular segments to $BC,AC,AB$ respectively.
Prove that : $ABC$~$A_{1}B_{1}C_{1}$
In triangle $ABC$ , $w$ is a circle which passes through $B,C$ and intersects $AB,AC$ at $E,F$ respectively. $BF,CE$ intersect the circumcircle of $ABC$ at $B',C'$ respectively. Let $A'$ be a point on $BC$ such that $\angle C'A'B=\angle B'A'C$ . Prove that if we change $w$, then all the circumcircles of triangles $A'B'C'$ passes through a common point.
Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively. Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. Prove that $XYII_a$ is cyclic.
Given triangle $\triangle ABC$ and let $D,E,F$ be the foot of angle bisectors of $A,B,C$ ,respectively. $M,N$ lie on $EF$ such that $AM=AN$. Let $H$ be the foot of $A$-altitude on $BC$.
Points $K,L$ lie on $EF$ such that triangles $\triangle AKL, \triangle HMN$ are correspondingly similiar (with the given order of vertices) such that $AK \not\parallel HM$ and $AK \not\parallel HN$. Show that: $DK=DL$
2017 Iran MO 3rd Round first geometry P2
Let $ABCD$ be a trapezoid ($AB<CD,AB\parallel CD$) and $P\equiv AD\cap BC$. Suppose that $Q$ be a point inside $ABCD$ such that $\angle QAB=\angle QDC=90-\angle BQC$. Prove that $\angle PQA=2\angle QCD$.
2017 Iran MO 3rd Round first geometry P3
Let $ABC$ be an acute-angle triangle. Suppose that $M$ be the midpoint of $BC$ and $H$ be the orthocenter of $ABC$. Let $F\equiv BH\cap AC$ and $E\equiv CH\cap AB$. Suppose that $X$ be a point on $EF$ such that $\angle XMH=\angle HAM$ and $A,X$ are in the distinct side of $MH$. Prove that $AH$ bisects $MX$.
Let $ABC$ be an acute-angle triangle. Suppose that $M$ be the midpoint of $BC$ and $H$ be the orthocenter of $ABC$. Let $F\equiv BH\cap AC$ and $E\equiv CH\cap AB$. Suppose that $X$ be a point on $EF$ such that $\angle XMH=\angle HAM$ and $A,X$ are in the distinct side of $MH$. Prove that $AH$ bisects $MX$.
2017 Iran MO 3rd Round finals geometry P1
Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.
Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.
2017 Iran MO 3rd Round finals geometry P2
Assume that $P$ be an arbitrary point inside of triangle $ABC$. $BP$ and $CP$ intersects $AC$ and $AB$ in $E$ and $F$, respectively. $EF$ intersects the circumcircle of $ABC$ in $B'$ and $C'$ (Point $E$ is between of $F$ and $B'$). Suppose that $B'P$ and $C'P$ intersects $BC$ in $C''$ and $B''$ respectively. Prove that $B'B''$ and $C'C''$ intersect each other on the circumcircle of $ABC$.
Assume that $P$ be an arbitrary point inside of triangle $ABC$. $BP$ and $CP$ intersects $AC$ and $AB$ in $E$ and $F$, respectively. $EF$ intersects the circumcircle of $ABC$ in $B'$ and $C'$ (Point $E$ is between of $F$ and $B'$). Suppose that $B'P$ and $C'P$ intersects $BC$ in $C''$ and $B''$ respectively. Prove that $B'B''$ and $C'C''$ intersect each other on the circumcircle of $ABC$.
In triangle $ABC$ points $P$ and $Q$ lies on the external bisector of $\angle A$ such that $B$ and $P$ lies on the same side of $AC$. Perpendicular from $P$ to $AB$ and $Q$ to $AC$ intersect at $X$. Points $P'$ and $Q'$ lies on $PB$ and $QC$ such that $PX=P'X$ and $QX=Q'X$. Point $T$ is the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$. Prove that $P',Q'$ and $T$ are collinear if and only if $\angle PBA+\angle QCA=90^{\circ}$.
Incircle of triangle $ABC$ is tangent to sides $BC,CA,AB$ at $D,E,F$,respectively.Points $P,Q$ are inside angle $BAC$ such that $FP=FB,FP||AC$ and $EQ=EC,EQ||AB$.Prove that $P,Q,D$ are collinear.
Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$)
Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.
2018 Iran MO 3rd Round first geometry P3
$H$ is the orthocenter of acude triangle $ABC$.Let $\omega$ be the circumcircle of $BHC$ with center $O'$.$\Omega$ is the nine-point circle of $ABC$.$X$ is an arbitrary point on arc $BHC$ of $\omega$ and $AX$ intersects $\Omega$ at $Y$.$P$ is a point on $\Omega$ such that $PX=PY$.Prove that $O'PX=90$.
2018 Iran MO 3rd Round first geometry P4
for acute triangle $\triangle ABC$ with orthocenter $H$, and $E,F$ the feet of altitudes for $B,C$, we have $P$ on $EF$ such as that $HO \perp HP$. $Q$ is on segment $AH$ so $HM \perp PQ$. prove $QA=3QH$
2019 Iran MO 3rd Round midterms geometry P1
Given a cyclic quadrilateral $ABCD$. There is a point $P$ on side $BC$ such that $\angle PAB=\angle PDC=90^\circ$. The medians of vertexes $A$ and $D$ in triangles $PAB$ and $PDC$ meet at $K$ and the bisectors of $\angle PAB$ and $\angle PDC$ meet at $L$. Prove that $KL\perp BC$.
Incircle of triangle $ABC$ is tangent to sides $BC,CA,AB$ at $D,E,F$,respectively.Points $P,Q$ are inside angle $BAC$ such that $FP=FB,FP||AC$ and $EQ=EC,EQ||AB$.Prove that $P,Q,D$ are collinear.
Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$)
Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.
2018 Iran MO 3rd Round first geometry P3
$H$ is the orthocenter of acude triangle $ABC$.Let $\omega$ be the circumcircle of $BHC$ with center $O'$.$\Omega$ is the nine-point circle of $ABC$.$X$ is an arbitrary point on arc $BHC$ of $\omega$ and $AX$ intersects $\Omega$ at $Y$.$P$ is a point on $\Omega$ such that $PX=PY$.Prove that $O'PX=90$.
2018 Iran MO 3rd Round first geometry P4
for acute triangle $\triangle ABC$ with orthocenter $H$, and $E,F$ the feet of altitudes for $B,C$, we have $P$ on $EF$ such as that $HO \perp HP$. $Q$ is on segment $AH$ so $HM \perp PQ$. prove $QA=3QH$
2019 Iran MO 3rd Round midterms geometry P1
Given a cyclic quadrilateral $ABCD$. There is a point $P$ on side $BC$ such that $\angle PAB=\angle PDC=90^\circ$. The medians of vertexes $A$ and $D$ in triangles $PAB$ and $PDC$ meet at $K$ and the bisectors of $\angle PAB$ and $\angle PDC$ meet at $L$. Prove that $KL\perp BC$.
2019 Iran MO 3rd Round midterms geometry P2
Consider an acute-angled triangle $ABC$ with $AB=AC$ and $\angle A>60^\circ$. Let $O$ be the circumcenter of $ABC$. Point $P$ lies on circumcircle of $BOC$ such that $BP\parallel AC$ and point $K$ lies on segment $AP$ such that $BK=BC$. Prove that $CK$ bisects the arc $BC$ of circumcircle of $BOC$.
Consider an acute-angled triangle $ABC$ with $AB=AC$ and $\angle A>60^\circ$. Let $O$ be the circumcenter of $ABC$. Point $P$ lies on circumcircle of $BOC$ such that $BP\parallel AC$ and point $K$ lies on segment $AP$ such that $BK=BC$. Prove that $CK$ bisects the arc $BC$ of circumcircle of $BOC$.
2019 Iran MO 3rd Round midterms geometry P3
Consider a triangle $ABC$ with circumcenter $O$ and incenter $I$. Incircle touches sides $BC,CA$ and $AB$ at $D, E$ and $F$. $K$ is a point such that $KF$ is tangent to circumcircle of $BFD$ and $KE$ is tangent to circumcircle of $CED$. Prove that $BC,OI$ and $AK$ are concurrent.
2019 Iran MO 3rd Round finals geometry P1
Consider a triangle $ABC$ with incenter $I$. Let $D$ be the intersection of $BI,AC$ and $CI$ intersects the circumcircle of $ABC$ at $M$. Point $K$ lies on the line $MD$ and $\angle KIA=90^\circ$. Let $F$ be the reflection of $B$ about $C$. Prove that $BIKF$ is cyclic.
2019 Iran MO 3rd Round finals geometry P2
In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.
2019 Iran MO 3rd Round finals geometry P1
Consider a triangle $ABC$ with incenter $I$. Let $D$ be the intersection of $BI,AC$ and $CI$ intersects the circumcircle of $ABC$ at $M$. Point $K$ lies on the line $MD$ and $\angle KIA=90^\circ$. Let $F$ be the reflection of $B$ about $C$. Prove that $BIKF$ is cyclic.
In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.
2019 Iran MO 3rd Round finals geometry P3
Given an inscribed pentagon $ABCDE$ with circumcircle $\Gamma$. Line $\ell$ passes through vertex $A$ and is tangent to $\Gamma$. Points $X,Y$ lie on $\ell$ so that $A$ lies between $X$ and $Y$. Circumcircle of triangle $XED$ intersects segment $AD$ at $Q$ and circumcircle of triangle $YBC$ intersects segment $AC$ at $P$. Lines $XE,YB$ intersects each other at $S$ and lines $XQ, Y P$ at $Z$. Prove that circumcircle of triangles $XY Z$ and $BES$ are tangent.
Given an inscribed pentagon $ABCDE$ with circumcircle $\Gamma$. Line $\ell$ passes through vertex $A$ and is tangent to $\Gamma$. Points $X,Y$ lie on $\ell$ so that $A$ lies between $X$ and $Y$. Circumcircle of triangle $XED$ intersects segment $AD$ at $Q$ and circumcircle of triangle $YBC$ intersects segment $AC$ at $P$. Lines $XE,YB$ intersects each other at $S$ and lines $XQ, Y P$ at $Z$. Prove that circumcircle of triangles $XY Z$ and $BES$ are tangent.
Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.
Triangle $ABC$ with it's circumcircle $\Gamma$ is given. Points $D$ and $E$ are chosen on segment $BC$ such that $\angle BAD=\angle CAE$. The circle $\omega$ is tangent to $AD$ at $A$ with it's circumcenter lies on $\Gamma$. Reflection of $A$ through $BC$ is $A'$. If the line $A'E$ meet $\omega$ at $L$ and $K$. Then prove either $BL$ and $CK$ or $BK$ and $CL$ meet on $\Gamma$.
The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.
Triangle $ABC$ is given. Let $O$ be it's circumcenter. Let $I$ be the center of it's incircle.The external angle bisector of $A$ meet $BC$ at $D$. And $I_A$ is the $A$-excenter . The point $K$ is chosen on the line $AI$ such that $AK=2AI$ and $A$ is closer to $K$ than $I$. If the segment $DF$ is the diameter of the circumcircle of triangle $DKI_A$, then prove $OF=3OI$.
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