Found problems: 320
2018 India PRMO, 5
Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.
2013 Dutch BxMO/EGMO TST, 5
Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre of $\triangle BCM$. Let $N$ be the second intersection pointof $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot |BN|$.
Geometry Mathley 2011-12, 9.1
Let $ABC$ be a triangle with $(O), (I)$ being the circumcircle, and incircle respectively. Let $(I)$ touch $BC,CA$, and $AB$ at $A_0, B_0, C_0$ let $BC,CA$, and $AB$ intersect $B_0C_0, C_0A_0, A_0Bv$ at $A_1, B_1$, and $C_1$ respectively. Prove that $OI$ passes through the orthocenter of four triangles $A_0B_0C_0, A_0B_1C_1, B_0C_1A_1,C_0A_1B_1$.
Nguyá»…n Minh HÃ
2010 Sharygin Geometry Olympiad, 4
In triangle $ABC$, touching points $A', B'$ of the incircle with $BC, AC$ and common point $G$ of segments $AA'$ and $BB'$ were marked. After this the triangle was erased. Restore it by the ruler and the compass.
2023 Belarus - Iran Friendly Competition, 4
Let $\Gamma$ be the incircle of a non-isosceles triangle $ABC$, $I$ be it’s incenter. Let $A_1,
B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$ respectively. Let $A_2 = \Gamma \cap AA_1$,
$M = C_1B_1 \cap AI$, $P$ and $Q$ be the other (different from $A_1$ and $A_2$) intersection points of $\Gamma$ and $A_1M$,
$A_2M$ respectively. Prove that $A$, $P$ and $Q$ are colinear.
2019 Sharygin Geometry Olympiad, 7
Let $P$ be an arbitrary point on side $BC$ of triangle $ABC$. Let $K$ be the incenter of triangle $PAB$. Let the incircle of triangle $PAC$ touch $BC$ at $F$. Point $G$ on $CK$ is such that $FG // PK$. Find the locus of $G$.
1970 IMO, 1
$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.
2009 Bulgaria National Olympiad, 2
In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.
2013 Poland - Second Round, 2
Circles $o_1$ and $o_2$ with centers in $O_1$ and $O_2$, respectively, intersect in two different points $A$ and $B$, wherein angle $O_1AO_2$ is obtuse. Line $O_1B$ intersects circle $o_2$ in point $C \neq B$. Line $O_2B$ intersects circle $o_1$ in point $D \neq B$. Show that point $B$ is incenter of triangle $ACD$.
2015 Thailand Mathematical Olympiad, 4
Let $\vartriangle ABC$ be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.
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 $ABC$.
1999 Israel Grosman Mathematical Olympiad, 3
For every triangle $ABC$, denote by $D(ABC)$ the triangle whose vertices are the tangency points of the incircle of $\vartriangle ABC$ with the sides. Assume that $\vartriangle ABC$ is not equilateral.
(a) Prove that $D(ABC)$ is also not equilateral.
(b) Find in the sequence $T_1 = \vartriangle ABC, T_{k+1} = D(T_k)$ for $k \in N$ a triangle whose largest angle $\alpha$ satisfies $0 < \alpha -60^o < 0.0001^o$
2010 Mathcenter Contest, 3
Let triangle $ABC$ be a triangle right at $B$. The inscribed circle is tangent to sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $CF$ intersect the circle at the point $P$. If $\angle APB=90^{\circ}$, find the value of $\dfrac{CP+CD}{PF}$.
[i](tatari/nightmare)[/i]
Kvant 2023, M2767
It is easy to prove that in a right triangle the sum of the radii of the incircle and three excircles is equal to the perimeter. Prove that the opposite statement is also true.
[i]Proposed by I. Weinstein[/i]
2019 Saudi Arabia Pre-TST + Training Tests, 1.3
Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.
2017 Sharygin Geometry Olympiad, P15
Let $ABC$ be an acute-angled triangle with incircle $\omega$ and incenter $I$. Let $\omega$ touch $AB, BC$ and $CA $ at points $D, E, F$ respectively. The circles $\omega_1$ and $\omega_2$ centered at $J_1$ and $J_2$ respectively are inscribed into A$DIF$ and $BDIE$. Let $J_1J_2$ intersect $AB$ at point $M$. Prove that $CD$ is perpendicular to $IM$.
2024 New Zealand MO, 6
Let $\omega$ be the incircle of scalene triangle $ABC$. Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$. Construct points $X^\prime$ and $Y^\prime$ on line segments $AB$ and $AC$ respectively such that $AX^\prime=XB$ and $AY^\prime=YC$. Let line $CX^\prime$ intersects $\omega$ at points $P,Q$ such that $P$ is closer to $C$ than $Q$. Also let $R^\prime$ be the intersection of lines $CX^\prime$ and $BY^\prime$. Prove that $CP=RX^\prime$.
2000 Czech and Slovak Match, 5
Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$. The incircle of the triangle $BCD$ touches $CD$ at $E$. Point $F$ is chosen on the bisector of the angle $DAC$ such that the lines $EF$ and $CD$ are perpendicular. The circumcircle of the triangle $ACF$ intersects the line $CD$ again at $G$. Prove that the triangle $AFG$ is isosceles.
VII Soros Olympiad 2000 - 01, 10.6
A circle is inscribed in triangle $ABC$. $M$ and $N$ are the points of its tangency with the sides $BC$ and $CA$, respectively. The segment $AM$ intersects $BN$ at point $P$ and the inscribed circle at point $Q$. It is known that $MP = a$, $PQ = b$. Find $AQ$.
2022 Turkey Team Selection Test, 8
$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that
$$|K_AL_A|=|K_BL_B|+|K_CL_C|$$
2019 Saudi Arabia IMO TST, 3
Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA, IB, IC$ cut $(d)$ at $A_1, B_1,C_1$ respectively. Suppose that $(d)$ cuts $BC, CA, AB$ at $M,N, P$ respectively. The lines through $M,N,P$ and respectively parallel to the internal bisectors of $A, B, C$ in triangle $ABC$ meet each other to define a triange $XYZ$. Prove that three lines $AA_1, BB_1, CC_1$ are concurrent and $IK$ is tangent to the circle $(XY Z)$
2000 Hungary-Israel Binational, 3
Let ${ABC}$ be a non-equilateral triangle. The incircle is tangent to the sides ${BC,CA,AB}$ at ${A_1,B_1,C_1}$, respectively, and M is the orthocenter of triangle ${A_1B_1C_1}$. Prove that ${M}$ lies on the line through the incenter and circumcenter of ${\vartriangle ABC}$.
2019 Paraguay Mathematical Olympiad, 5
A circle of radius $4$ is inscribed in a triangle $ABC$. We call $D$ the touchpoint between the circle and side BC. Let $CD =8$, $DB= 10$. What is the length of the sides $AB$ and $AC$?
KoMaL A Problems 2018/2019, A.748
The circles $\Omega$ and $\omega$ in its interior are fixed. The distinct points $A,B,C,D,E$ move on $\Omega$ in such a way that the line segments $AB,BC,CD,DE$ are tangents to $\omega$ .The lines $AB$ and $CD$ meet at point
$P$, the lines $BC$ and $DE$ meet at $Q$ . Let $R$ be the second intersection of the circles $BCP$and $CDQ$, other than $C$. Show that $R$ moves either on a circle or on a line.
2008 Germany Team Selection Test, 2
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.
[i]Author: Waldemar Pompe, Poland[/i]