Found problems: 1065
Prove that for any nonisosceles triangle $l_1^2>\sqrt3 S>l_2^2$, where $l_1, l_2$ are the greatest and the smallest bisectors of the triangle and $S$ is its area.
$ABCD$ is a cyclic quadrilateral. The lines $AD$ and $BC$ meet at X, and the lines $AB$ and $CD$ meet at $Y$ . The line joining the midpoints $M$ and $N$ of the diagonals $AC$ and $BD$, respectively, meets the internal bisector of angle $AXB$ at $P$ and the external bisector of angle $BYC$ at $Q$. Prove that $PXQY$ is a rectangle
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$.
[i]Albania[/i]
Let $ABCD$ be a convex quadrilateral and let $P$ be the point of intersection of $AC$ and $BD$. Suppose that $AC+AD=BC+BD$. Prove that the internal angle bisectors of $\angle ACB$, $\angle ADB$ and $\angle APB$ meet at a common point.
Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.
In the scalene triangle $ABC$,$\angle ACB=60$ and $\Omega$ is its cirumcirle.On the bisectors of the angles $BAC$ and $CBA$ points $A^\prime$,$B^\prime$ are chosen respectively such that $AB^\prime \parallel BC$ and $BA^\prime \parallel AC$.$A^\prime B^\prime$ intersects with $\Omega$ at $D,E$.Prove that triangle $CDE$ is isosceles.(A. Kuznetsov)
Two equal hard triangles are given. One of their angles is equal to $ \alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $ \alpha / 2$.
[i](No instruments are allowed, even a pencil.)[/i]
(E. Bakayev, A. Zaslavsky)
The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$.
[i]Proposed By Pedram Safaei[/i]
Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
In a convex quadrilateral $ABCD$ we draw the bisectors $\ell_a$, $\ell_b$, $\ell_c$, $\ell_d$ of external angles $A$, $B$, $C$, $D$ respectively. The intersection points of the lines $\ell_a$ and $\ell_b$, $\ell_b$ and $\ell_c$, $\ell_c$ and $\ell_d$, $\ell_d$ and $\ell_a$ are designated by $K$, $L$, $M$, $N$. It is known that $3$ perpendiculars drawn from $K$ on $AB$, from $L$ om $BC$, from $M$ on $CD$ intersect at one point. Prove that the quadrilateral $ABCD$ is cyclic.
Let $H$, $I$, $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$. When does the equality hold ?
Prove that the bisectors of the angles of a rectangle, extended to their mutual intersection, form a square.
Given triangle $ABC$, $AL$ bisects angle $\angle BAC$ with $L$ on side $BC$. Lines $LR$ and $LS$ are parallel to $BA$ and $CA$ respectively, $R$ on side $AC$ and$ S$ on side $AB$, respectively. Through point $B$ draw a perpendicular on $AL$, intersecting $LR$ at $M$. If point $D$ is the midpoint of $BC$, prove that that the three points $A, M, D$ lie on a straight line.
In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.
Triangle $ABC$ has side-lengths $AB=12$, $BC=24$, and $AC=18$. The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. What is the perimeter of $\triangle AMN$?
$ \textbf{(A)}\ 27 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 33 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 42
$
Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.
Let $BL, AD$ be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point $T$. It turned out that the altitude $LK$ of $\triangle ALB$ is divided in half by the line $AD$. Prove that $KT \perp BL$.
[i]Proposed by Mariia Rozhkova[/i]
Pentagon $ABCDE$ has circle $S$ inscribed into it. Side $BC$ is tangent to $S$ at point $K$. If $AB=BC=CD$, prove that angle $EKB$ is a right angle.
Let $AD$ be the inner angle bisector of the triangle $ABC$. The perpendicular on the side $BC$ at the point $D$ intersects the outer bisector of $\angle CAB$ at point $I$. The circle with center $I$ and radius $ID$ intersects the sides $AB$ and $AC$ at points $F$ and $E$ respectively. $A$-symmedian of $\Delta AFE$ intersects the circumcircle of $\Delta AFE$ again at point $X$. Prove that the circumcircles of $\Delta AFE$ and $\Delta BXC$ are tangent.
Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.
In the acute triangle $ABC$ the side $BC> AB$, and the angle bisector $BL = AB$. On the segment $BL$ there is a point $M$, for which $\angle AML = \angle BCA$. Prove that $AM = LC$.
Let $ABC$ be a triangle with $AB < AC$. The tangent to the circumcircle of $\triangle ABC$ at $A$ intersects $BC$ at $D$. The angle bisector of $\angle BAC$ intersect $BC$ at $E$. Suppose that the perpendicular bisector of $AE$ intersect $AB, AC$ at $P,Q$, respectively. Show that $$\sqrt{\frac{BP}{CQ}} = \frac{AC \cdot BD}{AB \cdot CD}$$
It is given triangle $ABC$. Let internal and external angle bisector of angle $\angle BAC$ intersect line $BC$ in points $D$ and $E$, respectively, and circumcircle of triangle $ADE$ intersects line $AC$ in point $F$. Prove that $FD$ is angle bisector of $\angle BFC$