Found problems: 320
2014 Iranian Geometry Olympiad (junior), P2
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
Kyiv City MO 1984-93 - geometry, 1990.8.2
A line passes through the center $O$ of an equilateral triangle $ABC$ and intersects the side $BC$. At what angle wrt $BC$ should this line be drawn this line so that its segment inside the triangle has the smallest possible length?
Ukraine Correspondence MO - geometry, 2016.7
The circle $\omega$ inscribed in an isosceles triangle $ABC$ ($AC = BC$) touches the side $BC$ at point $D$ .On the extensions of the segment $AB$ beyond points $A$ and $B$, respectively mark the points $K$ and $L$ so that $AK = BL$, The lines $KD$ and $LD$ intersect the circle $\omega$ for second time at points $G$ and $H$, respectively. Prove that point $A$ belongs to the line $GH$.
2024 Sharygin Geometry Olympiad, 17
Let $ABC$ be a non-isosceles triangle, $\omega$ be its incircle. Let $D, E, $ and $F$ be the points at which the incircle of $ABC$ touches the sides $BC, CA, $ and $AB$ respectively. Let $M$ be the point on ray $EF$ such that $EM = AB$. Let $N$ be the point on ray $FE$ such that $FN = AC$. Let the circumcircles of $\triangle BFM$ and $\triangle CEN$ intersect $\omega$ again at $S$ and $T$ respectively. Prove that $BS, CT, $ and $AD$ concur.
Mathley 2014-15, 3
Let the incircle $\gamma$ of triangle $ABC$ be tangent to $BA, BC$ at $D, E$, respectively. A tangent $t$ to $\gamma$ , distinct from the sidelines, intersects the line $AB$ at $M$. If lines $CM, DE$ meet at$ K$, prove that lines $AK,BC$ and $t$ are parallel or concurrent.
Michel Bataille , France
2014 Costa Rica - Final Round, 5
Let $ABC$ be a triangle, with $A'$, $B'$, and $C'$ the points of tangency of the incircle with $BC$, $CA$, and $AB$ respectively. Let $X$ be the intersection of the excircle with respect to $A$ with $AB$, and $M$ the midpoint of $BC$. Let $D$ be the intersection of $XM$ with $B'C'$. Show that $\angle C'A'D' = 90^o$.
1997 Bosnia and Herzegovina Team Selection Test, 4
$a)$ In triangle $ABC$ let $A_1$, $B_1$ and $C_1$ be touching points of incircle $ABC$ with $BA$, $CA$ and $AB$, respectively. Let $l_1$, $l_2$ and $l_3$ be lenghts of arcs $ B_1C_1$, $A_1C_1$, $B_1A_1$ of incircle $ABC$, respectively, which does not contain points $A_1$, $B_1$ and $C_1$, respectively.
Does the following inequality hold: $$ \frac{a}{l_1}+\frac{b}{l_2}+\frac{c}{l_3} \geq \frac{9\sqrt{3}}{\pi}$$
$b)$ Tetrahedron $ABCD$ has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides
2011 Ukraine Team Selection Test, 6
The circle $ \omega $ inscribed in triangle $ABC$ touches its sides $AB, BC, CA$ at points $K, L, M$ respectively. In the arc $KL$ of the circle $ \omega $ that does not contain the point $M$, we select point $S$. Denote by $P, Q, R, T$ the intersection points of straight $AS$ and $KM, ML$ and $SC, LP$ and $KQ, AQ$ and $PC$ respectively. It turned out that the points $R, S$ and $M$ are collinear. Prove that the point $T$ also lies on the line $SM$.
2012 Ukraine Team Selection Test, 4
Given an isosceles triangle $ABC$ ($AB = AC$), the inscribed circle $\omega$ touches its sides $AB$ and $AC$ at points $K$ and $L$, respectively. On the extension of the side of the base $BC$, towards $B$, an arbitrary point $M$. is chosen. Line $M$ intersects $\omega$ at the point $N$ for the second time, line $BN$ intersects the second point $\omega$ at the point $P$. On the line $PK$, there is a point $X$ such that $K$ lies between $P$ and $X$ and $KX = KM$. Determine the locus of the point $X$.
2021-IMOC, G11
The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.
2023 Sharygin Geometry Olympiad, 22
Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$, and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$. The tangent to the incircle at $P$ meets $BC$ at point $X$, and the tangent to the excircle at $Q$ meets $BC$ at $Y$. Prove that $MX=MY$.
1982 IMO Longlists, 36
A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.
2014 Thailand TSTST, 2
In a triangle $ABC$, the incircle with incenter $I$ is tangent to $BC$ at $A_1, CA$ at $B_1$, and $AB$ at $C_1$. Denote the intersection of $AA_1$ and $BB_1$ by $G$, the intersection of $AC$ and $A_1C_1$ by $X$, and the intersection of $BC$ and $B_1C_1$ by $Y$ . Prove that $IG \perp XY$ .
2022 Yasinsky Geometry Olympiad, 5
Point $X$ is chosen on side $AD$ of square $ABCD$. The inscribed circle of triangle $ABX$ touches $AX$, $BX$, and $AB$ at points $N$, $K$, and $F$, respectively. Prove that the ray $NK$ passes through the center $O$ of the square $ABCD$.
(Dmytro Shvetsov)
2013 Oral Moscow Geometry Olympiad, 6
The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.
2005 Sharygin Geometry Olympiad, 11.4
In the triangle $ABC , \angle A = \alpha, BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.
2021 Saudi Arabia Training Tests, 18
Let $ABC$ be a triangle with $AB < AC$ and incircle $(I)$ tangent to $BC$ at $D$. Take $K$ on $AD$ such that $CD = CK$. Suppose that $AD$ cuts $(I)$ at $G$ and $BG$ cuts $CK$ at $L$. Prove that K is the midpoint of $CL$.
2018 Bosnia And Herzegovina - Regional Olympiad, 4
Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$
Novosibirsk Oral Geo Oly IX, 2019.1
The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$.
[img]https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png[/img]
Swiss NMO - geometry, 2018.6
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 $BD = EF$.
1970 IMO Longlists, 39
$M$ 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 $r_1r_2q=rq_1q_2$.
I Soros Olympiad 1994-95 (Rus + Ukr), 9.6
In the triangle $ABC$, the orthocenter $H$ lies on the inscribed circle. Is this triangle necessarily isosceles?
2015 JBMO Shortlist, 5
Let $ABC$ be an acute triangle with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle touches the sides ${BC, CA}$ and ${AB}$ at ${D, E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$ onto ${BC}$ meets ${EF}$ at ${M}$, and similarly the perpendicular line erected at ${B}$ onto ${BC}$ meets ${EF}$ at ${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$, and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$.
Ruben Dario & Leo Giugiuc (Romania)
1998 Singapore Team Selection Test, 1
Let $I$ be the centre of the inscribed circle of the non-isosceles triangle $ABC$, and let the circle touch the sides $BC, CA, AB$ at the points $A_1, B_1, C_1$ respectively. Prove that the centres of the circumcircles of $\vartriangle AIA_1,\vartriangle BIB_1$ and $\vartriangle CIC_1$ are collinear.
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 $r_1r_2q=rq_1q_2$.