Found problems: 1389
2006 Czech and Slovak Olympiad III A, 4
Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.
2019 Yasinsky Geometry Olympiad, p2
A scalene triangle $ABC$ is given. It is known that $I$ is the center of the inscribed circle in this triangle, $D, E, F$ points are the touchpoints of this circle with the sides $AB, BC, CA$, respectively. Let $P$ be the intersection point of lines $DE$ and $AI$. Prove that $CP \perp AI$.
(Vtalsh Winds)
2017 Azerbaijan Team Selection Test, 2
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2017 Romanian Master of Mathematics Shortlist, G2
Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$.
proposed by Luis Eduardo Garcia Hernandez, Mexico
2021-IMOC, G9
Let the incenter and the $A$-excenter of $\triangle ABC$ be $I$ and $I_A$, respectively. Let $BI$ intersect $AC$ at $E$ and $CI$ intersect $AB$ at $F$. Suppose that the reflections of $I$ with respect to $EF$, $FI_A$, $EI_A$ are $X$, $Y$, $Z$, respectively. Show that $\odot(XYZ)$ and $\odot(ABC)$ are tangent to each other.
2010 Belarus Team Selection Test, 8.3
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
2016 Oral Moscow Geometry Olympiad, 1
The line passing through the center $I$ of the inscribed circle of a triangle $ABC$, perpendicular to $AI$ and intersects sides $AB$ and $AC$ at points $C'$ and $B'$, respectively. In the triangles $BC'I$ and $CB'I$, the altitudes $C'C_1$ and $B'B_1$ were drawn, respectively. Prove that the midpoint of the segment $B_1C_1$ lies on a straight line passing through point $I$ and perpendicular to $BC$.
2023 Girls in Mathematics Tournament, 4
Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.
2024 Sharygin Geometry Olympiad, 16
Let $AA_1, BB_1, $ and $CC_1$ be the bisectors of a triangle $ABC$. The segments $BB_1$ and $A_1C_1$ meet at point $D$. Let $E$ be the projection of $D$ to $AC$. Points $P$ and $Q$ on sides $AB$ and $BC$ respectively are such that $EP = PD, EQ = QD$. Prove that $\angle PDB_1 = \angle EDQ$.
2020 BMT Fall, 21
Let $\vartriangle ABC$ be a right triangle with legs $AB = 6$ and $AC = 8$. Let $I$ be the incenter of $\vartriangle ABC$ and $X$ be the other intersection of $AI$ with the circumcircle of $\vartriangle ABC$. Find $\overline{AI} \cdot \overline{IX}$.
1962 IMO, 5
On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.
1972 IMO Longlists, 12
A circle $k = (S, r)$ is given and a hexagon $AA'BB'CC'$ inscribed in it. The lengths of sides of the hexagon satisfy $AA' = A'B, BB' = B'C, CC' = C'A$. Prove that the area $P$ of triangle $ABC$ is not greater than the area $P'$ of triangle $A'B'C'$. When does $P = P'$ hold?
2002 All-Russian Olympiad Regional Round, 10.3
The perpendicular bisector to side $AC$ of triangle $ABC$ intersects side $BC$ at point $M$ (see fig.). The bisector of angle $\angle AMB$ intersects the circumcircle of triangle $ABC$ at point $K$. Prove that the line passing through the centers of the inscribed circles triangles $AKM$ and $BKM$, perpendicular to the bisector of angle $\angle AKB$.
[img]https://cdn.artofproblemsolving.com/attachments/b/4/b53ec7df0643a90b835f142d99c417a2a1dd45.png[/img]
2017 China Second Round Olympiad, 1
Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$.
2017 Taiwan TST Round 3, 2
Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$
2016 Saudi Arabia BMO TST, 2
Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly.
a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$.
b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.
2015 Iran MO (3rd round), 4
Let $ABC$ be a triangle with incenter $I$. Let $K$ be the midpoint of $AI$ and $BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N$. points $P,Q$ lie on $AM,AN$ respectively such that $\angle ABK=\angle PBC,\angle ACK=\angle QCB$. Prove that $P,Q,I$ are collinear.
2011 Paraguay Mathematical Olympiad, 5
In a rectangle triangle, let $I$ be its incenter and $G$ its geocenter. If $IG$ is parallel to one of the catheti and measures $10 cm$, find the lengths of the two catheti of the triangle.
2019 India IMO Training Camp, P1
In an acute angled triangle $ABC$ with $AB < AC$, let $I$ denote the incenter and $M$ the midpoint of side $BC$. The line through $A$ perpendicular to $AI$ intersects the tangent from $M$ to the incircle (different from line $BC$) at a point $P$> Show that $AI$ is tangent to the circumcircle of triangle $MIP$.
[i]Proposed by Tejaswi Navilarekallu[/i]
2022 India National Olympiad, 1
Let $D$ be an interior point on the side $BC$ of an acute-angled triangle $ABC$. Let the circumcircle of triangle $ADB$ intersect $AC$ again at $E(\ne A)$ and the circumcircle of triangle $ADC$ intersect $AB$ again at $F(\ne A)$. Let $AD$, $BE$, and $CF$ intersect the circumcircle of triangle $ABC$ again at $D_1(\ne A)$, $E_1(\ne B)$ and $F_1(\ne C)$, respectively. Let $I$ and $I_1$ be the incentres of triangles $DEF$ and $D_1E_1F_1$, respectively. Prove that $E,F, I, I_1$ are concyclic.
2023 Peru MO (ONEM), 4
Let $ABC$ be an acute scalene triangle and $K$ be a point inside it that belongs to the bisector of the angle $\angle ABC$. Let$ P$ be the point where the line $AK$ intersects the line perpendicular to $AB$ that passes through $B$, and let $Q$ be the point where the line $CK$ intersects the line perpendicular to $CB$ that passes through $B$. Let $L$ be the foot of the perpendicular drawn from $K$ on the line $AC$. Prove that if $P Q$ is perpendicular to $BL$, then $K$ is the incenter of $ABC$.
2017 Sharygin Geometry Olympiad, P5
A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of
а) the vertices of their greatest angles,
b) their incenters.
2021 Taiwan APMO Preliminary First Round, 2
(a) Let the incenter of $\triangle ABC$ be $I$. We connect $I$ other $3$ vertices and divide $\triangle ABC$ into $3$ small triangles which has area $2,3$ and $4$. Find the area of the inscribed circle of $\triangle ABC$.
(b) Let $ABCD$ be a parallelogram. Point $E,F$ is on $AB,BC$ respectively. If $[AED]=7,[EBF]=3,[CDF]=6$, then find $[DEF].$ (Here $[XYZ]$ denotes the area of $XYZ$)
2021-IMOC, G10
Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.
Croatia MO (HMO) - geometry, 2018.7
Given an acute-angled triangle $ABC$ in which $|AB| <|AC|$. Point $D$ is the midpoint of the shorter arc $BC$ of its circumcircle. The point $I$ is the center of its incircle, and the point $J$ is symmetric point of $I$ wrt line $BC$. The line $DJ$ intersects the circumcircle of the triangle $ABC$ at the point $E$ belonging to the arc $AB$. Prove that $|AI |= |IE|$.