Found problems: 320
2015 Oral Moscow Geometry Olympiad, 4
In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.
1995 IMO Shortlist, 3
The incircle of triangle $ \triangle ABC$ touches the sides $ BC$, $ CA$, $ AB$ at $ D, E, F$ respectively. $ X$ is a point inside triangle of $ \triangle ABC$ such that the incircle of triangle $ \triangle XBC$ touches $ BC$ at $ D$, and touches $ CX$ and $ XB$ at $ Y$ and $ Z$ respectively.
Show that $ E, F, Z, Y$ are concyclic.
2018 Portugal MO, 2
In the figure, $[ABCD]$ is a square of side $1$. The points $E, F, G$ and $H$ are such that $[AFB], [BGC], [CHD]$ and $[DEA]$ are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square $[EFGH]$ has all the same radius, what is the measure of the radius of the circles?
[img]https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png[/img]
2020 CHKMO, 3
Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.
2017 Saudi Arabia IMO TST, 2
Let $ABCD$ be the circumscribed quadrilateral with the incircle $(I)$. The circle $(I)$ touches $AB, BC, C D, DA$ at $M, N, P,Q$ respectively. Let $K$ and $L$ be the circumcenters of the triangles $AMN$ and $APQ$ respectively. The line $KL$ cuts the line $BD$ at $R$. The line $AI$ cuts the line $MQ$ at $J$. Prove that $RA = RJ$.
2017 Ukrainian Geometry Olympiad, 4
In the right triangle $ABC$ with hypotenuse $AB$, the incircle touches $BC$ and $AC$ at points ${{A}_{1}}$ and ${{B}_{1}}$ respectively. The straight line containing the midline of $\Delta ABC$ , parallel to $AB$, intersects its circumcircle at points $P$ and $T$. Prove that points $P,T,{{A}_{1}}$ and ${{B}_{1}}$ lie on one circle.
Croatia MO (HMO) - geometry, 2012.7
Let the points $M$ and $N$ be the intersections of the inscribed circle of the right-angled triangle $ABC$, with sides $AB$ and $CA$ respectively , and points $P$ and $Q$ respectively be the intersections of the ex-scribed circles opposite to vertices $B$ and $C$ with direction $BC$. Prove that the quadrilateral $MNPQ$ is a cyclic if and only if the triangle $ABC$ is right-angled with a right angle at the vertex $A$.
2016 Middle European Mathematical Olympiad, 6
Let $ABC$ be a triangle for which $AB \neq AC$. Points $K$, $L$, $M$ are the midpoints of the sides $BC$, $CA$, $AB$.
The incircle of $ABC$ with center $I$ is tangent to $BC$ in $D$. A line passing through the midpoint of $ID$ perpendicular to $IK$ meets the line $LM$ in $P$.
Prove that $\angle PIA = 90 ^{\circ}$.
Geometry Mathley 2011-12, 7.3
Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in AD$, $P,Q \in BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral.
Nguyễn Văn Linh
2016 Saint Petersburg Mathematical Olympiad, 3
The circle inscribed in the triangle $ABC$ is tangent to side $AC$ at point $B_1$, and to side $BC$ at point $A_1$. On the side $AB$ there is a point $K$ such that $AK = KB_1, BK = KA_1$. Prove that $ \angle ACB\ge 60$
2006 Sharygin Geometry Olympiad, 15
A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$ at points $A_1,B_1,C_1$, respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$, and $M$ is the midpoint of the segment $PA_1$. Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.
Mathley 2014-15, 6
Let the inscribed circle $(I)$ of the triangle $ABC$, touches $CA, AB$ at $E, F$. $P$ moves along $EF$, $PB$ cuts $CA$ at $M, MI$ cuts the line, through $C$ perpendicular to $AC$, at $N$. Prove that the line through $N$ is perpendicular to $PC$ crosses a fixed point as $P$ moves.
Tran Quang Hung, High School of Natural Sciences, Hanoi National University
Geometry Mathley 2011-12, 10.1
Let $ABC$ be a triangle with two angles $B,C$ not having the same measure, $I$ be its incircle, $(O)$ its circumcircle. Circle $(O_b)$ touches $BA,BC$ and is internally tangent to $(O)$ at $B_1$. Circle $(O_c)$ touches $CA,CB$ and is internally tangent to $(O)$ at $C_1$. Let $S$ be the intersection of $BC$ and $B_1C_1$. Prove that $\angle AIS = 90^o$.
Nguyễn Minh Hà
Durer Math Competition CD Finals - geometry, 2017.D+5
The inscribed circle of the triangle $ABC$ touches the sides $BC, CA, AB$ at points $A_1, B_1, C_1$ respectively. The points $P_b, Q_b, R_b$ are the points of the segments $BC_1, C_1A_1, A_1B$, respectively, such that $BP_bQ_bR_b$ is parallelogram. In the same way, the points $P_c, Q_c, R_c$ are the points of the sections $CB_1, B_1A_1, A_1C$, respectively such that $CP_cQ_cR_c$ is a parallelogram. The intersection of the lines $P_bR_b$ and $P_cR_c$ is $T$. Show that $TQ_b = TQ_c$.
Kyiv City MO Juniors 2003+ geometry, 2007.9.3
On a straight line $4$ points are successively set , $A, P, Q,W $, which are the points of intersection of the bisector $AL $ of the triangle $ABC$ with the circumscribed and inscribed circle. Knowing only these points, construct a triangle $ABC $.
2009 Peru MO (ONEM), 2
In a quadrilateral $ABCD$, a circle is inscribed that is tangent to the sides $AB, BC, CD$ and $DA$ at points $M, N, P$ and $Q$, respectively. If $(AM) (CP) = (BN) (DQ)$, prove that $ABCD$ is an cyclic quadrilateral.
2018 Switzerland - Final Round, 6
Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.
2009 Postal Coaching, 5
Let $ABCD$ be a quadrilateral that has an incircle with centre $O$ and radius $r$. Let $P = AB \cap CD$, $Q = AD \cap BC$, $E = AC \cap BD$. Show that $OE \cdot d = r^2$, where $d$ is the distance of $O$ from $PQ$.
2013 Silk Road, 2
Circle with center $I$, inscribed in a triangle $ABC$ , touches the sides $BC$ and $AC$ at points $A_1$ and $B_1$ respectively. On rays $A_1I$ and $B_1I$, respectively, let be the points $A_2$ and $B_2$ such that $IA_2=IB_2=R$, where $R$is the radius of the circumscribed circle of the triangle $ABC$. Prove that:
a) $AA_2 = BB_2 = OI$ where $O$ is the center of the circumscribed circle of the triangle $ABC$,
b) lines $AA_2$ and $BB_2$ intersect on the circumcircle of the triangle $ABC$.
2014 India PRMO, 16
In a triangle $ABC$, let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$, respectively. If $\angle BAC = 40^o$, what is the value of $\angle QPR$ in degrees ?
2016 All-Russian Olympiad, 7
In triangle $ABC$,$AB<AC$ and $\omega$ is incirle.The $A$-excircle is tangent to $BC$ at $A^\prime$.Point $X$ lies on $AA^\prime$ such that segment $A^\prime X$ doesn't intersect with $\omega$.The tangents from $X$ to $\omega$ intersect with $BC$ at $Y,Z$.Prove that the sum $XY+XZ$ not depends to point $X$.(Mitrofanov)
2013 Czech And Slovak Olympiad IIIA, 3
In the parallelolgram A$BCD$ with the center $S$, let $O$ be the center of the circle of the inscribed triangle $ABD$ and let $T$ be the touch point with the diagonal $BD$. Prove that the lines $OS$ and $CT$ are parallel.
2010 Balkan MO Shortlist, G3
The incircle of a triangle $A_0B_0C_0$ touches the sides $B_0C_0,C_0A_0,A_0B_0$ at the points $A,B,C$ respectively, and the incircle of the triangle $ABC$ with incenter $ I$ touches the sides $BC,CA, AB$ at the points $A_1, B_1,C_1$, respectively. Let $\sigma(ABC)$ and $\sigma(A_1B_1C)$ be the areas of the triangles $ABC$ and $A_1B_1C$ respectively. Show that if $\sigma(ABC) = 2 \sigma(A_1B_1C)$ , then the lines $AA_0, BB_0,IC_1$ pass through a common point .
2011 Sharygin Geometry Olympiad, 11
The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.
2022 239 Open Mathematical Olympiad, 2
Point $I{}$ is the center of the circle inscribed in the quadrilateral $ABCD$. Prove that there is a point $K{}$ on the ray $CI$ such that $\angle KBI=\angle KDI=\angle BAI$.