geometry problems from Iranian Geometry Olympiad (IGO) [all authors / proposers]
with aops links in the names s
by Mahan Tajrobekar
by Mahdi Etesami Fard
by Ali Zooelm
by Mahdi Etesami Fard
by
Morteza Saghafian
by Mahdi Etesami Fard
by Morteza Saghafian
by Davood
Vakili
by Mahdi Etesami Fard
by Morteza Saghafian
by Davood Vakili
by Ali Golmakani
by Alexey Zaslavsky (Russia)
by Morteza Saghafian
by Morteza Saghafian
by Nikolai Beluhov (Bulgaria) and Morteza Saghafian
by Davood Vakili
by Iman Maghsoudi
by Boris Frenkin (Russia)
Convex hexagon $A_1A_2A_3A_4A_5A_6$ lies in the interior of convex hexagon $B_1B_2B_3B_4B_5B_6$ such that $A_1A_2 \parallel B_1B_2$, $A_2A_3 \parallel B_2B_3$,..., $A_6A_1 \parallel B_6B_1$. Prove that the areas of simple hexagons $A_1B_2A_3B_4A_5B_6$ and $B_1A_2B_3A_4B_5A_6$ are equal. (A simple hexagon is a hexagon which does not intersect itself.)
There are three rectangles in the following figure. The lengths of some segments are shown.
Find the length of the segment $XY$ .
with aops links in the names s
Iranian Geometry Olympiad problems 2014-2017 ΕΝ in pdf
Iranian Geometry Olympiad all 2014-2021 in pdf with solutions
Iranian Geometry Olympiad all 2014-2021 in pdf with solutions
2014 - 2022
IGO 2014 Junior 1 / Senior 1
In a right triangle ABC we have <A = 90o, <C = 30o. Denote by C the circle passing through A which is tangent to BC at the midpoint. Assume that C intersects AC and the circumcircle of ABC at N and M respectively. Prove that MN $\perp$ BC.
by Mahdi Etesami Fard
IGO 2014 Junior 2
The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$
by Mahdi Etesami Fard
Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$.
by Morteza Saghafian
In a triangle ABC we have $\angle C = \angle A + 90^o$. The point $D$ on the continuation of $BC$ is given such that $AC = AD$. A point $E$ in the side of $BC$ in which $A$ doesn’t lie is chosen such that $\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A$ . Prove that $\angle CED = \angle ABC$.
by Morteza Saghafian
Two points $X, Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$ . Let $M$ denotes the midpoint of the chord $AX$ . Show that $BM +CM > AY$ .
In a right triangle ABC we
have <A = 90o, <C = 30o. Denote by C the circle passing through A which is tangent to BC
at the midpoint. Assume that C intersects
AC and the circumcircle of ABC at N and M respectively.
Prove that MN $\perp$ BC.
by Mahdi Etesami Fard
In a quadrilateral ABCD we
have <B = <D = 60o. Consider the line
which is drawn from M, the
midpoint of AD, parallel to CD. Assume this line intersects BC at P. A point X lies
on CD such that BX = CX. Prove that AB =
BP
<MXB = 60o.
by Davood Vakili
An acute-angled triangle ABC is
given. The circle with diameter BC intersects
AB, AC at E, F respectively. Let M be the midpoint of BC and P the intersection point of AM and EF. X is a point on the arc EF and Y the second intersection point of XP with circle mentioned above. Show that <XAY = <XYM.
by Ali Zooelm
The tangent line to circumcircle of the acute-angled triangle ABC (AC > AB) at A intersects
the continuation of BC at P. We denote by O the circumcenter of ABC. X is a point OP such
that <AXP = 90o. Two points E, F respectively
on AB, AC at the same side of OP
are chosen such that <EXP =
<ACX, <FXO = <ABX. If K, L denote the intersection points of EF with the circumcircle of △ABC, show that OP is tangent to the circumcircle of △KLX.
by Mahdi Etesami Fard
Two points P, Q lie on the side BC of triangle ABC and have the same distance to the
midpoint. The perpendiculars from P,
Q tp BC intesect AC, AB at E, F respectively.
Let M be the intersection point
of PF and EQ. If H1 and H2
denote the orthocenter of △BFP and △CEQ respectively, show that AM $\perp$ H1H2.
by Mahdi Etesami Fard
Suppose that $I$ is incenter of $\vartriangle ABC$ and $CI$ inresects $AB$ at $D$.In circumcircle of $\vartriangle ABC$, $T$ is midpoint of arc $BAC$ and $BI$ intersect this circle at $M$. If $MD$ intersects $AT$ at $N$, prove that: $BM \parallel CN$.
by Ali Zooelm
We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof)
A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example:
A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example:
by Mahdi Etesami Fard
Let ABC be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.
by Mahdi Etesami Fard
In the figure below, we know that $AB = CD$ and $BC = 2AD$. Prove that $<BAD = 30^o$.
In rectangle $ABCD$, the points $M,N,P,Q$ lie on $AB,BC,CD,DA$ respectively such that the area of triangles $AQM,BMN,CNP,DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram.
Do there exist $6$ circles in the plane such that every circle passes through centers of exactly $3$ other circles?
In the figure below, the points P,A,B lie on a circle. The point Q lies
inside the circle such that <PAQ = 90o and PQ = BQ. Prove that
the value of <AQB - <PQA is equal to the arc AB.
In acute-angled triangle ABC, BH is the altitude of the vertex B. The
points D and E are midpoints of AB and AC respectively. Suppose that F be the
reflection of H with respect to ED. Prove that the line BF passes through
circumcenter of ABC.
by Davood Vakili
In triangle ABC, the points M,N,K are the midpoints of BC,CA,AB respectively.
Let ωB and ωC be two semicircles with diameter AC and AB respectively, outside the
triangle. Suppose that MK and MN intersect ωC and ωB at X and Y respectively. Let the tangents at X and Y to ωC and ωB respectively, intersect at Z. prove that AZ $\perp$ BC.
by Mahdi Etesami Fard
Let ABC be an equilateral triangle with circumcircle ω and circumcenter O. Let P be the point on the arc BC (the arc which A
doesn't lie ). Tangent to ω at P intersects
extensions of AB and AC at K and L respectively. Show that <KOL > 90o.
by Iman Maghsoudi
a) Do there exist 5 circles in the plane such that every circle passes
through centers of exactly 3 circles?
b) Do there exist 6 circles in the plane such that every circle passes
through centers of exactly 3 circles?
by Morteza Saghafian
Two circles ω1 and ω2 (with centers O1 and O2
respectively) intersect at A and B. The point X lies on ω2. Let point Y be a point on ω1 such that <XBY = 90o. Let X΄ be the second point of intersection of the line O1X and ω2 and K be the second point of intersection of X΄Y and ω2. Prove that X is the midpoint of arc AK.
by Davood Vakili
Let ABC be an equilateral triangle with circumcircle ω and circumcenter O. Let P be the point on the arc BC (the arc which A
doesn't lie ). Tangent to ω at P intersects extensions of AB and AC at K and L respectively. Show that
<KOL > 90o.
by Iman Maghsoudi
Let H be the orthocenter of the triangle ABC. Let l1 and l2
be two lines passing through H and perpendicular to each other. l1
intersects BC and extension of AB at D and Z respectively, and l2
intersects BC and extension of AC at E and X respectively. Let Y be a point
such that YD//AC and YE//AB. Prove that X,Y,Z are collinear.
by Ali Golmakani
In triangle ABC, we draw the circle with center A and radius AB. This
circle intersects AC at two points. Also we draw the circle with center A and
radius AC and this circle intersects AB at two points. Denote these four points
by A1,A2,A3,A4. Find the points B1,B2,B3,B4
and C1,C2,C3,C4 similarly. Suppose
that these 12 points lie on two circles. Prove that the triangle ABC is
isosceles.
by Morteza Saghafian
Rectangles ABA1B2, BCB1C2,
CAC1A2 lie outside triangle ABC. Let C΄ be a point such that C΄A1 $\perp$
A1C2 and C΄B2 $\perp$ B2C1.
Points A΄ and B΄ are defined similarly. Prove
that lines AA΄, BB΄, CC΄ concur.
by Alexey Zaslavsky (Russia)
Ali wants to move from point $A$ to point $B$. He cannot walk inside the black
areas but he is free to move in any direction inside the white areas (not only
the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and write its length.
by Morteza Saghafian
Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$.
by Iman Maghsoudi
Suppose that $ABCD$ is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these $4$ new points, there is only one point inside the quadrilateral $ABCD$.
In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$.
by Mahdi Etesami Fard
Let ABCD be a convex quadrilateral with these
properties: <ADC = 135 o and <ADB - <ABD = 2<DAB = 4<CBD. If BC = √2 CD , prove that AB =
BC + AD.
by Mahdi Etesami Fard
In trapezoid ABCD with AB // CD, ω1 and ω2 are two circles with diameters AD and BC, respectively. Let X and Y be
two arbitrary points on ω1 and ω2, respectively. Show that the length of segment XY is
not more than half of the perimeter of ABCD.
by Mahdi Etesami Fard
Let two circles C1 and C2 intersect in points A
and B. The tangent to C1 at A intersects C2 in P and the
line PB intersects C1 for the second time in Q (suppose that Q is
outside C2). The tangent to C2 from Q intersects C1
and C2 in C and D, respectively (The points A and D lie on different
sides of the line PQ). Show that AD is bisector of the angle <CAP.
by Iman Maghsoudi
Find all positive integers N such that there exists a triangle which can
be dissected into N similar quadrilaterals.
by Nikolai Beluhov (Bulgaria) and Morteza Saghafian
Let ω be the circumcircle of right-angled triangle ABC (<A = 90o).
Tangent to ω at point A intersects the line BC in point P. Suppose that M is the midpoint
of (the smaller) arc AB, and PM intersects ω for the second time in Q. Tangent to ω at point Q intersects AC in K. Prove that <PKC = 90o.
by Davood Vakili
Let the circles ω and ω΄ intersect in points A and B.
Tangent to circle ω at A intersects ω΄ in C and tangent to circle ω΄ at A intersects ω in D. Suppose that the internal bisector of <CAD intersects ω and ω΄ at E and F, respectively,
and the external bisector of <CAD intersects ω and ω΄ in X and Y , respectively.
Prove that the perpendicular bisector of XY is tangent to the circumcircle of
triangle BEF.
by Mahdi Etesami Fard
Let the circles ω and ω΄ intersect in A and B. Tangent to circle ω at A intersects ω΄ in C and tangent to circle ω΄ at A intersects ω in D. Suppose that the segment CD intersects ω and ω in E and F, respectively
(assume that E is between F and C). The perpendicular to AC from E intersects ω΄ in point P and perpendicular
to AD from F intersects ω in point Q (The
points A, P and Q lie on the same side of the line CD). Prove that the points
A, P and Q are collinear.
by Mahdi Etesami Fard
In acute-angled triangle ABC, altitude of A meets BC at D, and M is midpoint
of AC. Suppose that X is a point such that <AXB = <DXM = 90o (assume
that X and C lie on opposite sides of the line BM). Show that <XMB = 2<MBC.
by Davood Vakili
Let P be the intersection point of sides AD and BC of a convex quadrilateral
ABCD. Suppose that I1 and I2 are the incenters of
triangles PAB and PDC, respectively. Let O be the circumcenter of PAB, and H
the orthocenter of PDC. Show that the circumcircles of triangles AI1B
and DHC are tangent together if and only if the circumcircles of triangles AOB
and DI2C are tangent together.
by Hooman Fattahimoghaddam
In a convex quadrilateral ABCD, the lines AB and CD meet at point E and
the lines AD and BC meet at point F. Let P be the intersection point of
diagonals AC and BD. Suppose that ω1 is a circle passing through D and tangent to AC at P. Also suppose that
ω2 is a circle passing through C and tangent to BD at P.
Let X be the intersection point of ω1 and AD, and Y be the intersection point of ω2 and BC. Suppose that the circles ω1 and ω2 intersect each other in Q for the second time. Prove that the
perpendicular from P to the line EF passes through the circumcenter of triangle
XQY
by Iman Maghsoudi
Do there exist six points X1,X2,Y1,Y2,Z1,Z2
in the plane such that all of the triangles XiYjZk
are similar for 1 ≤ i, j, k ≤ 2 ?
by Morteza Saghafian
Each side of square ABCD with side length of 4 is divided into equal
parts by three points. Choose one of the three points from each side, and
connect the points consecutively to obtain a quadrilateral. Which numbers can
be the area of this quadrilateral? Just write the numbers without proof.
by Hirad
Aalipanah
Find the angles of triangle ABC.
by Morteza Saghafian
In the regular pentagon ABCDE, the perpendicular at C to CD meets AB at F.
Prove that AE + AF = BE.
by Alireza Cheraghi
P1, P2, ... , P100 are 100 points on the plane, no three of them are
collinear. For each three points, call their triangle clockwise if the increasing
order of them is in clockwise order. Can the number of clockwise triangles be
exactly 2017?
by Morteza Saghafian
In the isosceles triangle ABC (AB = AC), let l be a line parallel to BC through
A. Let D be an arbitrary point on l.
Let E, F be the feet of perpendiculars through A to BD, CD respectively.
Suppose that P, Q are the images of E, F on l.
Prove that AP + AQ ≤ AB.
by Morteza Saghafian
Let ABC be an acute-angled triangle with A = 60o. Let E, F be
the feet of altitudes through B, C respectively. Prove that CE - BF = 3 / 2 (AC -
AB).
by Fatemeh Sajadi
Two circles ω1, ω2 intersect at A, B. An arbitrary line
through B meets ω1, ω2 at
C, D respectively. The points E, F are chosen on ω1, ω2 respectively so that CE = CB, BD = DF.
Suppose that BF meets ω1at P, and BE meets ω2 at Q. Prove that A, P, Q are collinear.
by Iman Maghsoudi
On the plane, n points are given (n > 2). No three of them are
collinear. Through each two of them the line is drawn, and among the other
given points, the one nearest to this line is marked (in each case this point
occurred to be unique). What is the maximal possible number of marked points
for each given n?
by Boris Frenkin (Russia)
In the isosceles triangle ABC (AB = AC), let l be a line parallel to BC through A. Let D be an arbitrary point
on l. Let E, F be the feet of
perpendiculars through A to BD, CD respectively. Suppose that P, Q are the
images of E, F on l. Prove that AP + AQ
≤ AB.
by Morteza Saghafian
Let X, Y be two points on the side BC of triangle ABC such that 2XY = BC.
(X is between B, Y ) Let AA΄ be the diameter of the circumcircle of triangle AXY . Let P be the point
where AX meets the perpendicular from B to BC, and Q be the point where AY meets
the perpendicular from C to BC. Prove that the tangent line from A΄ to the circumcircle of AXY passes
through the circumcenter of triangle APQ.
by Iman Maghsoudi
In triangle ABC, the incircle, with center I, touches the side BC at
point D. Line DI meets AC at X. The tangent line from X to the incircle (different
from AC) intersects AB at Y . If YI and BC intersect at point Z, prove that AB =
BZ.
by Hooman Fattahimoghaddam
We have six pairwise non-intersecting circles that the radius of each is
at least one. Prove that the radius of any circle intersecting all the six
circles, is at least one.
by Mohammad Ali Abam - Morteza Saghafian
Let O be the circumcenter of triangle ABC. Line CO intersects the
altitude through A at point K. Let P, M be the midpoints of AK, AC respectively.
If PO intersects BC at Y , and the circumcircle of triangle BCM meets AB at X,
prove that BXOY is cyclic.
by Ali Daeinabi - Hamid Pardazi
Three circles ω1, ω2, ω3 are tangent to
line l at points A, B, C (B lies
between A, C) and ω2 is externally tangent to the other two. Let X, Y
be the intersection points of ω2 with the other common external
tangent of ω1, ω3. The perpendicular line through B to l meets ω2 again at Z. Prove
that the circle with diameter AC touches ZX, ZY .
by Iman Maghsoudi - Siamak Ahmadpour
Sphere S touches a plane. Let A, B, C, D be four points on this plane
such that no three of them are collinear. Consider the point A΄ such that S is
tangent to the faces of tetrahedron A΄BCD. Points B΄, C΄, D΄ are defined similarly. Prove that A΄, B΄, C΄, D΄ are coplanar
and the plane A΄B΄C΄D΄ touches S.
by Alexey Zaslavsky (Russia)
As shown below, there is a $40\times30$ paper with a filled $10\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is to minimize the total length of the straight cuts. How to achieve this goal, and what is that minimized length? Show the correct cuts and write the final answer. There is no need to prove the answer.
by Morteza Saghafian
by Hirad Aalipanah and Mahdi Etesamifard
by Morteza SaghafianIn the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$.
by Mahdi Etesamifard
There are two circles with centers $O_1,O_2$ lie inside of circle $\omega$ and are tangent to it. Chord $AB$ of $\omega$ is tangent to these two circles such that they lie on opposite sides of this chord. Prove that $\angle O_1AO_2 + \angle O_1BO_2 > 90^\circ$.
by Iman Maghsoudi
There are some segments on the plane such that no two of them intersect each other (even at the ending points). We say segment $AB$ breaks segment $CD$ if the extension of $AB$ cuts $CD$ at some point between $C$ and $D$.
a) Is it possible that each segment when extended from both ends, breaks exactly one other segment from each way?
b) A segment is called surrounded if from both sides of it, there is exactly one segment that breaks it.
(e.g. segment $AB$ in the figure.) Is it possible to have all segments to be surrounded?
by Morteza Saghafian
Find the length of the segment $XY$ .
by Hirad Aalipanah
In convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ meet at the point $P$. We know that $\angle DAC = 90^o$ and $2 \angle ADB = \angle ACB$. If we have $ \angle DBC + 2 \angle ADC = 180^o$ prove that $2AP = BP$.
by Iman Maghsoudi
Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ .
by Alireza Dadgarnia
We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.)
by Mahdi Etesamifard and Morteza Saghafian
Suppose that $ABCD$ is a parallelogram such that $\angle DAC = 90^o$. Let $H$ be the foot of perpendicular from $A$ to $DC$, also let $P$ be a point along the line $AC$ such that the line $PD$ is tangent to the circumcircle of the triangle $ABD$. Prove that $\angle PBA = \angle DBH$.
by Iman Maghsoudi
Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$. Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q \in \omega_2$. An arbitrary point $X$ lies on $\omega_1$. Line $AX$ intersects $ \omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$. Line $Y'B$ intersects $ \omega_1$ for the second time at $X'$. Prove that $PX = PX'$.
In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$.
by Morteza Saghafian
by Fatemeh Sajadi
Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.
by Mahdi Etesamifard
Quadrilateral $ABCD$ is circumscribed around a circle. Diagonals $AC,BD$ are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments $AB,BC,CD$ and $DA$ at points $K,L,M$ and $N$. Given that $KLMN$ is cyclic, prove that so is $ABCD$.
by Nikolai Beluhov (Bulgaria)
$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$. Another circle passing through $C,D$ is tangent to $AB$ at point $F$. Point $G$ is the intersection point of $AE,DF$, and point $H$ is the intersection point of $BE,CF$. Prove that the incenters of triangles $AGF,BHF,CHE,DGE$ lie on a circle.
by Le Viet An (Vietnam)
There is a table in the shape of a $8\times 5$ rectangle with four holes on its corners. After shooting a ball from points $A, B$ and $C$ on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.)
by Hirad Alipanah
As shown in the figure, there are two rectangles $ABCD$ and $PQRD$ with the same area, and with parallel corresponding edges. Let points $N,$ $M$ and $T$ be the midpoints of segments $QR,$ $PC$ and $AB$, respectively. Prove that points $N,M$ and $T$ lie on the same line.
Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$ and $C$, respectively. Prove that quadrilateral $XPYC$ is a parallelogram.
Circles $\omega_1$ and $\omega_2$ intersect each other at points $A$ and $B$. Point $C$ lies on the tangent line from $A$ to $\omega_1$ such that $\angle ABC = 90^\circ$. Arbitrary line $\ell$ passes through $C$ and cuts $\omega_2$ at points $P$ and $Q$. Lines $AP$ and $AQ$ cut $\omega_1$ for the second time at points $X$ and $Z$ respectively. Let $Y$ be the foot of altitude from $A$ to $\ell$. Prove that points $X, Y$ and $Z$ are collinear.
by Morteza Saghafian
There are $n>2$ lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all?
by Boris Frenkin (Russia)
Quadrilateral $ABCD$ is given such that $\angle DAC = \angle CAB = 60^\circ,$ and $AB = BD - AC.$ Lines $AB$ and $CD$ intersect each other at point $E$. Prove that $ \angle ADB = 2\angle BEC.$
by Iman Maghsoudi
For a convex polygon (i.e. all angles less than $180^\circ$) call a diagonal bisector if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon?
by Morteza Saghafian
by Iman Maghsoudi
Find all quadrilaterals $ABCD$ such that all four triangles $DAB$, $CDA$, $BCD$ and $ABC$ are similar to one-another.
by Morteza Saghafian
Three circles $\omega_1$, $\omega_2$ and $\omega_3$ pass through one common point, say $P$. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$, $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent.
by Mahdi Etesamifard
Let $ABCD$ be a parallelogram and let $K$ be a point on line $AD$ such that $BK=AB$. Suppose that $P$ is an arbitrary point on $AB$, and the perpendicular bisector of $PC$ intersects the circumcircle of triangle $APD$ at points $X$, $Y$. Prove that the circumcircle of triangle $ABK$ passes through the orthocenter of triangle $AXY$.
by Iman Maghsoudi
Let $ABC$ be a triangle with $\angle A = 60^\circ$. Points $E$ and $F$ are the foot of angle bisectors of vertices $B$ and $C$ respectively. Points $P$ and $Q$ are considered such that quadrilaterals $BFPE$ and $CEQF$ are parallelograms. Prove that $\angle PAQ > 150^\circ$. (Consider the angle $PAQ$ that does not contain side $AB$ of the triangle.)
by Alireza Dadgarnia
by Iman Maghsoudi
Is it true that in any convex $n$-gon with $n > 3$, there exists a vertex and a diagonal passing through this vertex such that the angles of this diagonal with both sides adjacent to this vertex are acute?
by Boris Frenkin (Russia)
Circles $\omega_1$ and $\omega_2$ have centres $O_1$ and $O_2$, respectively. These two circles intersect at points $X$ and $Y$. $AB$ is common tangent line of these two circles such that $A$ lies on $\omega_1$ and $B$ lies on $\omega_2$. Let tangents to $\omega_1$ and $\omega_2$ at $X$ intersect $O_1O_2$ at points $K$ and $L$, respectively. Suppose that line $BL$ intersects $\omega_2$ for the second time at $M$ and line $AK$ intersects $\omega_1$ for the second time at $N$. Prove that lines $AM, BN$ and $O_1O_2$ concur.
by Dominik Burek (Poland)
Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$. $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of arc $BC$ of $\Gamma$ (the one that doesn't contain $A$). $X$ and $Y$ are points on $\Gamma$ such that $BX\parallel CY\parallel AM$. Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$. Let $\omega$ be the circumcircle of triangle $ZMN$. Line $AM$ meets $\omega$ for the second time at $P$. Let $K$ be a point on $\omega$ such that $KN\parallel AM$, $\omega_b$ be a circle that passes through $B$, $X$ and tangents to $BC$ and $\omega_c$ be a circle that passes through $C$, $Y$ and tangents to $BC$. Prove that circle with center $K$ and radius $KP$ is tangent to 3 circles $\omega_b$, $\omega_c$ and $\Gamma$.
by Tran Quan (Vietnam)
Let points $A, B$ and $C$ lie on the parabola $\Delta$ such that the point $H$, orthocenter of triangle $ABC$, coincides with the focus of parabola $\Delta$. Prove that by changing the position of points $A, B$ and $C$ on $\Delta$ so that the orthocenter remain at $H$, inradius of triangle $ABC$ remains unchanged.
by Mahdi Etesamifard
By a fold of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper.
Start with this shaded polygon and make a rectangle-shaped paper from it with at most 5 number
of folds. Describe your solution by introducing the folding lines and drawing the shape after each fold on your solution sheet.
(Note that the folding lines do not have to coincide with the grid lines of the shape.)
by Mahdi Etesamifard
A parallelogram $ABCD$ is given ($AB \neq BC$). Points $E$ and $G$ are chosen on the line $\overline{CD}$ such that $\overline{AC}$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$. The line $\overline{BC}$ intersects $\overline{AE}$ and $\overline{AG}$ at $F$ and $H$, respectively. Prove that the line $\overline{FG}$ passes through the midpoint of $HE$.
by Mahdi Etesamifard
According to the figure, three equilateral triangles with side lengths $a,b,c$ have one
common vertex and do not have any other common point. The lengths $x, y$, and $z$ are defined as
in the figure. Prove that $3(x+y+z)>2(a+b+c)$.
by Mahdi Etesamifard
Let $P$ be an arbitrary point in the interior of triangle $\triangle ABC$. Lines$\overline{BP}$ and $\overline{CP}$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$, respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$, respectively. Let the lines through $L$ and $K$ parallel to $\overline{CF}$ and $\overline{BE}$ intersect $\overline{BC}$ at $S$ and $T$, respectively; moreover, denote by $M$ and $N$ the reflection of $S$ and $T$ over the points $L$ and $K$, respectively. Prove that as $P$ moves in the interior of triangle $\triangle ABC$, line $\overline{MN}$ passes through a fixed point.
by Ali Zamani
We say two vertices of a simple polygon are visible from each other if either they are adjacent, or the segment joining them is completely inside the polygon (except two endpoints that lie on the boundary). Find all positive integers $n$ such that there exists a simple polygon with $n$ vertices in which every vertex is visible from exactly $4$ other vertices.
(A simple polygon is a polygon without hole that does not intersect itself.)
by Morteza Saghafian
A trapezoid $ABCD$ is given where $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the segment $AB$. Point $N$ is located on the segment $CD$ such that $\angle ADN = \frac{1}{2} \angle MNC$ and $\angle BCN = \frac{1}{2} \angle MND$. Prove that $N$ is the midpoint of the segment $CD$.
by Alireza Dadgarnia
Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$.
by Ali Zamani
In acute-angled triangle $ABC$ ($AC > AB$), point $H$ is the orthocenter and point $M$ is the midpoint of the segment $BC$. The median $AM$ intersects the circumcircle of triangle $ABC$ at $X$. The line $CH$ intersects the perpendicular bisector of $BC$ at $E$ and the circumcircle of the triangle $ABC$ again at $F$. Point $J$ lies on circle $\omega$, passing through $X, E,$ and $F$, such that $BCHJ$ is a trapezoid ($CB \parallel HJ$). Prove that $JB$ and $EM$ meet on $\omega$.
by Alireza Dadgarnia
Triangle $ABC$ is given. An arbitrary circle with center $J$, passing through $B$ and $C$, intersects the sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $X$ be a point such that triangle $FXB$ is similar to triangle $EJC$ (with the same order) and the points $X$ and $C$ lie on the same side of the line $AB$. Similarly, let $Y$ be a point such that triangle $EYC$ is similar to triangle $FJB$ (with the same order) and the points $Y$ and $B$ lie on the same side of the line $AC$. Prove that the line $XY$ passes through the orthocenter of the triangle $ABC$.
by Nguyen Van Linh (Vietnam)
Find all numbers $n \geq 4$ such that there exists a convex polyhedron with exactly $n$ faces, whose all faces are right-angled triangles.
(Note that the angle between any pair of adjacent faces in a convex polyhedron is less than $180^\circ$.)
by Hesam Rajabzadeh
Let $M,N,P$ be midpoints of $BC,AC$ and $AB$ of triangle $\triangle ABC$ respectively. $E$ and $F$ are two points on the segment $\overline{BC}$ so that $\angle NEC = \frac{1}{2} \angle AMB$ and $\angle PFB = \frac{1}{2} \angle AMC$. Prove that $AE=AF$.
by Alireza Dadgarnia
Let $\triangle ABC$ be an acute-angled triangle with its incenter $I$. Suppose that $N$ is the midpoint of the arc $BAC$ of the circumcircle of triangle $\triangle ABC$, and $P$ is a point such that $ABPC$ is a parallelogram.Let $Q$ be the reflection of $A$ over $N$ and $R$ the projection of $A$ on $\overline{QI}$. Show that the line $\overline{AI}$ is tangent to the circumcircle of triangle $\triangle PQR$
by Patrik Bak (Slovakia)
Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii.
(A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.)
Note. Weaker results with $2\sqrt{2}$ replaced by some other $c$ may be awarded points depending on the value of $c>2\sqrt{2}$
by Morteza Saghafian
Convex circumscribed quadrilateral $ABCD$ with its incenter $I$ is given such that its incircle is tangent to $\overline{AD},\overline{DC},\overline{CB},$ and $\overline{BA}$ at $K,L,M,$ and $N$. Lines $\overline{AD}$ and $\overline{BC}$ meet at $E$ and lines $\overline{AB}$ and $\overline{CD}$ meet at $F$. Let $\overline{KM}$ intersects $\overline{AB}$ and $\overline{CD}$ at $X,Y$, respectively. Let $\overline{LN}$ intersects $\overline{AD}$ and $\overline{BC}$ at $Z,T$, respectively. Prove that the circumcircle of triangle $\triangle XFY$ and the circle with diameter $EI$ are tangent if and only if the circumcircle of triangle $\triangle TEZ$ and the circle with diameter $FI$ are tangent.
by Mahdi Etesamifard
Consider an acute-angled triangle $\triangle ABC$ ($AC>AB$) with its orthocenter $H$ and circumcircle $\Gamma$.Points $M$,$P$ are midpoints of $BC$ and $AH$ respectively.The line $\overline{AM}$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $\overline{BC}$ so that $\overline{NX}$ is tangent to $\Gamma$.
Points $J$ and $K$ lie on the circle with diameter $MP$ such that $\angle AJP=\angle HNM$ ($B$ and $J$ lie one the same side of $\overline{AH}$) and circle $\omega_1$, passing through $K,H$, and $J$, and circle $\omega_2$ passing through $K,M$, and $N$, are externally tangent to each other. Prove that the common external tangents of $\omega_1$ and $\omega_2$ meet on the line $\overline{NH}$.
by Alireza Dadgarnia
With putting the four shapes drawn in the following figure together make a shape with at least two
reflection symmetries.
by Mahdi Etesamifard
Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that
the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$
is parallel to some side of $ABCD$.
by Josef Tkadlec (Czech Republic)
As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$,
$BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair
$(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be
bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$.
If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.
by Mahdi Etesamifard
In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$
in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the
reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles
$ADF$ and $BXY$ are concentric.
by Iman Maghsoudi
Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and
$$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which
by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$
(assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add
up to $90^o$.
by Morteza Saghafian
Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point $E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that $BE \perp HD$.
by Tran Quang Hung (Vietnam)
Let $ABCD$ be a parallelogram. Points $E, F$ lie on the sides $AB, CD$ respectively, such that $\angle EDC = \angle FBC$ and $\angle ECD = \angle FAD$. Prove that $AB \geq 2BC$.
by Pouria Mahmoudkhan Shirazi
Given a convex quadrilateral $ABCD$ with $AB = BC $and $\angle ABD = \angle BCD = 90$.Let point $E$ be the intersection of diagonals $AC$ and $BD$. Point $F$ lies on the side $AD$ such that $\frac{AF}{F D}=\frac{CE}{EA}$.. Circle $\omega$ with diameter $DF$ and the circumcircle of triangle $ABF$ intersect for the second time at point $K$. Point $L$ is the second intersection of $EF$ and $\omega$. Prove that the line $KL$ passes through the midpoint of $CE$.
by Mahdi Etesamifard and Amir Parsa Hosseini
Let $ABC$ be a scalene acute-angled triangle with its incenter $I$ and circumcircle $\Gamma$. Line $AI$ intersects $\Gamma$ for the second time at $M$. Let $N$ be the midpoint of $BC$ and $T$ be the point on $\Gamma$ such that $IN \perp MT$. Finally, let $P $ and $Q$ be the intersection points of $TB $ and $TC$, respectively, with the line perpendicular to $AI$ at $I$. Show that $PB = CQ$.
by Patrik Bak (Slovakia)
Consider a convex pentagon $ABCDE$ and a variable point $X$ on its side $CD$. Suppose that points $K, L$ lie on the segment $AX$ such that $AB = BK$ and $AE = EL$ and that the circumcircles of triangles $CXK$ and $DXL$ intersect for the second time at $Y$ . As $X$ varies, prove that all such lines $XY$ pass through a fixed point, or they are all parallel.
by Josef Tkadlec (Czech Republic)
Acute-angled triangle $ABC$ with circumcircle $\omega$ is given. Let $D$ be the midpoint of $AC$, $E$ be the foot of altitude from $A$ to $BC$, and $F$ be the intersection point of $AB$ and $DE$. Point $H$ lies on the arc $BC$ of $\omega$ (the one that does not contain $A$) such that $\angle BHE=\angle ABC$. Prove that $\angle BHF=90^\circ$.
by Harris Leung (Hong Kong)
Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.
by Tak Wing Ching (Hong Kong)
Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.
by Mahdi Etesamifard
$2021$ points on the plane in the convex position, no three collinear and no four concyclic, are given. Prove that there exist two of them such that every circle passing through these two points contains at least $673$ of the other points in its interior. (A finite set of points on the plane are in convex position if the points are the vertices of a convex polygon.)
by Morteza Saghafian
Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.
by Le Viet An (Vietnam)
Find the angles of the pentagon $ABCDE$ in the figure below.
by Morteza Saghafian
An isosceles trapezoid $ABCD$ $(AB \parallel CD)$ is given. Points $E$ and $F$ lie on the sides $BC$ and $AD$, and the points $M$ and $N$ lie on the segment $EF$ such that $DF = BE$ and $FM = NE$. Let $K$ and $L$ be the foot of perpendicular lines from $M$ and $N$ to $AB$ and $CD$, respectively. Prove that $EKFL$ is a parallelogram.
by Mahdi Etesamifard
Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$.
by Josef Tkadlec (Czech Republic)
Let $AD$ be the internal angle bisector of triangle $ABC$. The incircles of triangles $ABC$ and $ACD$ touch each other externally. Prove that $\angle ABC > 120^{\circ}$. (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.)
by Volodymyr Brayman (Ukraine)
a) Do there exist four equilateral triangles in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them?
b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them?
(Note that in both parts, there is no assumption on the intersection of interior of polygons.)
by Hesam Rajabzadeh
In the figure below we have $AX = BY$ . Prove that $ \angle XDA = \angle CDY$ .
by Iman Maghsoudi
Two circles $\omega_1$ and $\omega_2$ with equal radius intersect at two points $E$ and $X$. Arbitrary points $C, D$ lie on $\omega_1, \omega_2$. Parallel lines to $XC, XD$ from $E$ intersect $\omega_2, \omega_1$ at $A, B$, respectively. Suppose that $CD$ intersect $\omega_1, \omega_2$ again at $P, Q$, respectively. Prove that $ABPQ$ is cyclic.
by Ali Zamani
Let $O$ be the circumcenter of triangle $ABC$. Arbitrary points $M$ and $N$ lie on the sides $AC$ and $BC$, respectively. Points $P$ and $Q$ lie in the same half-plane as point $C$ with respect to the line $MN$, and satisfy $\triangle CMN \sim \triangle PAN \sim \triangle QMB$ (in this exact order). Prove that $OP=OQ$.
by Medeubek Kungozhin (Kazakhstan)
We call two simple polygons $P, Q$ $\textit{compatible}$ if there exists a positive integer $k$ such that each of $P, Q$ can be partitioned into $k$ congruent polygons similar to the other one. Prove that for every two even integers $m, n \geq 4$, there are two compatible polygons with $m$ and $n$ sides. (A simple polygon is a polygon that does not intersect itself.)
by Hesam Rajabzadeh
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ with center $O$. Let $P$ be the intersection of two diagonals $AC$ and $BD$. Let $Q$ be a point lying on the segment $OP$. Let $E$ and $F$ be the orthogonal projections of $Q$ on the lines $AD$ and $BC$, respectively. The points $M$ and $N$ lie on the circumcircle of triangle $QEF$ such that $QM \parallel AC$ and $QN \parallel BD$. Prove that the two lines $ME$ and $NF$ meet on the perpendicular bisector of segment $CD$.
by Tran Quang Hung (Vietnam)
Four points $A$, $B$, $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$. The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$. The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$. Prove that $MA=ML$.
by Mahdi Etesamifard
We are given an acute triangle $ABC$ with $AB\neq AC$. Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$, respectively, and let $M$ be the midpoints $EF$. Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$.
by Patrik Bak (Slovakia)
In triangle $ABC$ $(\angle A\neq 90^\circ)$, let $O$, $H$ be the circumcenter and the foot of the altitude from $A$ respectively. Suppose $M$, $N$ are the midpoints of $BC$, $AH$ respectively. Let $D$ be the intersection of $AO$ and $BC$ and let $H'$ be the reflection of $H$ about $M$. Suppose that the circumcircle of $OH'D$ intersects the circumcircle of $BOC$ at $E$. Prove that $NO$ and $AE$ are concurrent on the circumcircle of $BOC$.
by Mehran Talaei
Let $ABCD$ be a trapezoid with $AB\parallel CD$. Its diagonals intersect at a point $P$. The line passing through $P$ parallel to $AB$ intersects $AD$ and $BC$ at $Q$ and $R$, respectively. Exterior angle bisectors of angles $DBA$, $DCA$ intersect at $X$. Let $S$ be the foot of $X$ onto $BC$. Prove that if quadrilaterals $ABPQ$, $CDQP$ are circumcribed, then $PR=PS$.
by Dominik Burek (Poland)
Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$. Points $E$, $F$ lie on its side $AC$, $AB$, respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$, $S$ be the intersections of $EF$ with the shorter arcs $AB$, $AC$ of $\omega$, respectively. Suppose $K$, $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$, respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$, respectively, such that $PK$ and $QL$ are perpendicular to $BC$. Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$.
by Mehran Talaei
source: igo-official.ir
these are good questions
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