Found problems: 847
Let $ABC$ be a triangle with incentre $I$. The incircle of the triangle $ABC$ touches the sides $AC$ and $AB$ at points $E$ and $F$ respectively. Let $\ell_B$ and $\ell_C$ be the tangents to the circumcircle of $BIC$ at $B$ and $C$ respectively. Show that there is a circle tangent to $EF, \ell_B$ and $\ell_C$ with centre on the line $BC$.
[i]Proposed by Navneel Singhal, India[/i]
Given triangle $ABC$ and its incircle $\omega$ prove you can use just a ruler and drawing at most 8 lines to construct points$A',B',C'$ on $\omega$ such that $A,B',C'$ and $B,C',A'$ and $C,A',B'$ are collinear.
Given a triangle $ABC$ isosceles at $A.$ A point $P$ lying inside the triangle such that $\angle PBC=\angle PCA$ and let $M$ be the midpoint of $BC.$
Prove that: $\angle APB+ \angle MPC =180^{\circ}.$
Let $ABC$ be an acute-angled triangle with $AB<AC<BC$ and $c(O,R)$ the circumscribed circle. Let $D, E$ be points in the small arcs $AC, AB$ respectively. Let $K$ be the intersection point of $BD,CE$ and $N$ the second common point of the circumscribed circles of the triangles $BKE$ and $CKD$. Prove that $A, K, N$ are collinear if and only if $K$ belongs to the symmedian of $ABC$ passing from $A$.
Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$, respectively, are externally tangent at point $\lambda$. A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$, $O_2$ lie inside $C$. The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$. If $D$ is the midpoint of the segment $KL$, show that $\angle O_1OO_2 = \angle ADB$.
[i]Greece[/i]
$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.
In $\triangle ABC$, $AB > AC$. Point $P$ is on line $BC$ such that $AP$ is tangent to its circumcircle. Let $M$ be the midpoint of $AB$, and suppose the circumcircle of $\triangle PMA$ meets line $AC$ again at $N$. Point $Q$ is the reflection of $P$ with respect to the midpoint of segment $BC$. The line through $B$ parallel to $QN$ meets $PN$ at $D$, and the line through $P$ parallel to $DM$ meets the circumcircle of $\triangle PMB$ again at $E$. Show that the lines $PM$, $BE$, and $AC$ are concurrent.
Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.
Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.
Point $D$ is selected inside acute $\triangle ABC$ so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^{\circ} + \angle BAC$. Point $E$ is chosen on ray $BD$ so that $AE = EC$. Let $M$ be the midpoint of $BC$.
Show that line $AB$ is tangent to the circumcircle of triangle $BEM$.
[i]Proposed by Anton Trygub[/i]
Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and $BC$ at points $D$, $E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK||BC$.
Let $ABC$ be a triangle and let $D\in (BC)$ be the foot of the $A$- altitude. The circle $w$ with the diameter $[AD]$
meet again the lines $AB$ , $AC$ in the points $K\in (AB)$ , $L\in (AC)$ respectively. Denote the meetpoint $M$
of the tangents to the circle $w$ in the points $K$ , $L$ . Prove that the ray $[AM$ is the $A$-median in $\triangle ABC$ ([b][u]Serbia[/u][/b]).
A tangent line to circumcircle of acute triangle $ABC$ ($AC>AB$) at $A$ intersects with the extension of $BC$ at $P$. $O$ is the circumcenter of triangle $ABC$.Point $X$ lying on $OP$ such that $\measuredangle AXP=90^\circ$.Points $E$ and $F$ lying on $AB$ and $AC$,respectively,and they are in one side of line $OP$ such that $ \measuredangle EXP=\measuredangle ACX $ and $\measuredangle FXO=\measuredangle ABX $.
$K$,$L$ are points of intersection $EF$ with circumcircle of triangle $ABC$.prove that $OP$ is tangent to circumcircle of triangle $KLX$.
Author:Mehdi E'tesami Fard , Iran
A rectangle is said to be $ inscribed$ in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.
a) (version for grades 10-11)
Let $P$ be a point lying in the interior of a triangle. Show that the product of the distances from $P$ to the sides of the triangle is at least $8$ times less than the product of the distances from $P$ to the tangents to the circumcircle at the vertices of the triangle.
b) (version for grades 8-9)
Is it true that for any triangle there exists a point $P$ for which equality in the inequality from a) holds?
A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.
Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.
Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$. The incircle of triangle $ABC$ is tangent to the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D,E,F$ respectively. Let $M$ be the midpoint of $\overline{EF}$. Let $P$ be the projection of $A$ onto $BC$ and let $K$ be the intersection of $MP$ and $AD$. Prove that the circumcircles of triangles $AFE$ and $PDK$ have equal radius.
[i]Kyprianos-Iason Prodromidis[/i]
Let quadrilateral $A_1A_2A_3A_4$ be not cyclic and haves edges not parallel to each other.
Denote $B_i$ as the intersection of the tangent line at $A_i$ with respect to circle $A_{i-1}A_iA_{i+1}$ and the $A_{i+2}$-symmedian with respect to triangle $A_{i+1}A_{i+2}A_{i+3}$ and $C_i$ as the intersection of lines $A_iA_{i+1}$ and $B_iB_{i+1}$, where all indexes taken cyclically.
Prove that $C_1$, $C_2$, $C_3$, and $C_4$ are collinear.
Let $ABC$ be a right-angled triangle ($\angle C = 90^\circ$) and $D$ be the midpoint of an altitude from C. The reflections of the line $AB$ about $AD$ and $BD$, respectively, meet at point $F$. Find the ratio $S_{ABF}:S_{ABC}$.
Note: $S_{\alpha}$ means the area of $\alpha$.
The incircle of an acute-angled triangle $ABC$ touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, B_2$ are the midpoints of the segments $B_1C_1, A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
Let $ABC$ be a triangle with $AB > AC$ with incenter $I{}$. The internal bisector of the angle $BAC$ intersects the $BC$ at the point $D{}$. Let $M{}$ the midpoint of the segment $AD{}$, and let $F{}$ be the second intersection point of $MB$ with the circumcircle of the triangle $BIC$. Prove that $AF$ is perpendicular to $FC$.
Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$
a) Prove that $EF=\frac{BC}{2}.$
b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.