Found problems: 320
2009 Sharygin Geometry Olympiad, 8
A triangle $ABC$ is given, in which the segment $BC$ touches the incircle and the corresponding excircle in points $M$ and $N$. If $\angle BAC = 2 \angle MAN$, show that $BC = 2MN$.
(N.Beluhov)
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]
2001 Estonia Team Selection Test, 6
Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.
2004 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be a triangle inscribed in the circle $K$ and consider a point $M$ on the arc $BC$ that do not contain $A$. The tangents from $M$ to the incircle of $ABC$ intersect the circle $K$ at the points $N$ and $P$. Prove that if $\angle BAC = \angle NMP$, then triangles $ABC$ and $MNP$ are congruent.
Valentin Vornicu
[hide= about Romania JBMO TST 2004 in aops]I found the Romania JBMO TST 2004 links [url=https://artofproblemsolving.com/community/c6h5462p17656]here [/url] but they were inactive. So I am asking for solution for the only geo I couldn't find using search. The problems were found [url=https://artofproblemsolving.com/community/c6h5135p16284]here[/url].[/hide]
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.
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 $BC$.
2014 Junior Balkan Team Selection Tests - Romania, 4
In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC], A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the intersection point of the tangents in $A$ and $C $ to the circumcircle of triangle $ABC, X$ be the intersection point of lines $ZA$ and $A_1C_1$ and $Y$ be the intersection point of lines $ZC$ and $A_1C_1$.
a) Prove that $T$ is the incircle of triangle $XYZ$.
b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$.
c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.
Geometry Mathley 2011-12, 15.1
Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$.
Nguyễn Minh Hà
2019 India PRMO, 28
Let $ABC$ be a triangle with sides $51, 52, 53$. Let $\Omega$ denote the incircle of $\bigtriangleup ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$.
2017 Singapore MO Open, 1
The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.
2016 Sharygin Geometry Olympiad, P9
Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively, $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.
2016 ELMO Problems, 6
Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$.
(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$.
(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$.
[i]James Lin[/i]
Kharkiv City MO Seniors - geometry, 2021.10.5
The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.
2021-IMOC, G7
The incircle of triangle $ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let the tangents of $E$, $F$ with respect to $\odot(AEF)$ intersect at $P$, and $X$ be a point on $BC$ such that $EF$, $DP$, $AX$ are concurrent. Define $Q$, $Y$ and $R$, $Z$ similarly. Show that $X$, $Y$, $Z$ are collinear.
1998 Belarus Team Selection Test, 2
The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$.
I. Voronovich
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.
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 $A$.
Geometry Mathley 2011-12, 4.3
Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that
i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency.
ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$.
Nguyễn Minh Hà
2013 Thailand Mathematical Olympiad, 12
Let $\omega$ be the incircle of $\vartriangle ABC$, $\omega$ is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$ again at $P$. Lines $AP$ and $BC$ intersect at $M$. Let $N$ be a point on segment $AC$ so that $AE = CN$. Line $BN$ intersects $\omega$ at $Q$ (closer to $B$) and intersect $AM$ at $R$. Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$.
2008 Mathcenter Contest, 2
In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$.
[i](tatari/nightmare)[/i]
2010 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$ the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second intersection point of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.
2007 Silk Road, 2
Let $\omega$ be the incircle of triangle $ABC$ touches $BC$ at point $K$ . Draw a circle passing through points $B$ and $C$ , and touching $\omega$ at the point $S$ . Prove that $S K$ passes through the center of the exscribed circle of triangle $A B C$ , tangent to side $B C$ .
2024 Israel National Olympiad (Gillis), P6
Quadrilateral $ABCD$ is inscribed in a circle. Let $\omega_A$, $\omega_B$, $\omega_C$, $\omega_D$ be the incircles of triangles $DAB$, $ABC$, $BCD$, $CDA$ respectively. The common external common tangent of $\omega_A$, $\omega_B$, different from line $AB$, meets the external common tangent of $\omega_A$, $\omega_D$, different from $AD$, at point $A'$. Similarly, the external common tangent of $\omega_B$, $\omega_C$ different from $BC$ meets the external common tangent of $\omega_C$, $\omega_D$ different from $CD$ at $C'$.
Prove that $AA'\parallel CC'$.
2009 Switzerland - Final Round, 5
Let $ABC$ be a triangle with $AB \ne AC$ and incenter $I$. The incircle touches $BC$ at $D$. Let $M$ be the midpoint of $BC$ . Show that the line $IM$ bisects segment $AD$ .
Croatia MO (HMO) - geometry, 2016.7
Let $P$ be a point inside a triangle $ABC$ such that
$$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$
Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.