Found problems: 351
On a isosceles triangle $\triangle ABC$ with $AB=BC$ let $K,M$ be the midpoints of $AB,AC$ respectivily. Let $(CKB)$ intersect $BM$ at $N \ne M$, the line through $N$ parallel to $AC$ intersects $(ABC)$ at $A_1,C_1$. Show that $\triangle A_1BC_1$ is equilateral.
Let $ABC$ be a non isosceles triangle inscribed in a circle $(O)$ and $BE, CF$ are two angle bisectors intersect at $I$ with $E$ belongs to segment $AC$ and $F$ belongs to segment $AB$. Suppose that $BE, CF$ intersect $(O)$ at $M,N$ respectively. The line $d_1$ passes through $M$ and perpendicular to $BM$ intersects $(O)$ at the second point $P,$ the line $d_2$ passes through $N$ and perpendicular to $CN$ intersect $(O)$ at the second point $Q$. Denote $H, K$ are two midpoints of $MP$ and $NQ$ respectively.
1. Prove that triangles $IEF$ and $OKH$ are similar.
2. Suppose that S is the intersection of two lines $d_1$ and $d_2$. Prove that $SO$ is perpendicular to $EF$.
The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.
An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$. Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot |CG|$.
The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles
The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that $O$ is the centre of the circumscribed circle.
[u]other formulation:[/u]
$P$ is a convex polygon and $X$ is an interior point such that for every pair of vertices $A, B$, the triangle $XAB$ is isosceles. Prove that all the vertices of $P$ lie on a circle with center $X$.
Given the circle $c(O,R)$ (with center $O$ and radius $R$), one diameter $AB$ and midpoint $C$ of the arc $AB$. Consider circle $c_1(K,KO)$, where center $K$ lies on the segment $OA$, and consider the tangents $CD,CO$ from the point $C$ to circle $c_1(K,KO)$. Line $KD$ intersects circle $c(O,R)$ at points $E$ and $Z$ (point $E$ lies on the semicircle that lies also point $C$). Lines $EC$ and $CZ$ intersects $AB$ at points $N$ and $M$ respectively. Prove that quadrilateral $EMZN$ is an isosceles trapezoid, inscribed in a circle whose center lie on circle $c(O,R)$.
Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.
In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.
In an isosceles trapezoid $ABCD$ with bases $AD$ and $BC$, diagonals intersect at point $P$, and lines $AB$ and $CD$ intersect at point $Q$. $O_1$ and $O_2$ are the centers of the circles circumscribed around the triangles $ABP$ and $CDP$, $r$ is the radius of these circles. Construct the trapezoid ABCD given the segments $O_1O_2$, $PQ$ and radius $r$.
Consider an isosceles triangle $ABC$ with $AB = AC$. Let $\omega(XYZ)$ be the circumcircle of the triangle $XY Z$. The tangents to $\omega(ABC)$ through $B$ and $C$ meet at the point $D$. The point $F$ is marked on the arc $AB$ of $\omega(ABC)$ that does not contain $C$. Let $K$ be the point of intersection of lines $AF$ and $BD$ and $L$ the point of intersection of the lines $AB$ and $CF$. Let $T$ and $S$ be the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively. Suppose that the circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other at the point $P$. Prove that $P$ belongs to the line $AB$.
Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .
Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$ be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$.
Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.
Given an isosceles triangle $ABC$ with base $AB$. Point $K$ is taken on the extension of the side $AC$ (beyond the point $C$ ) so that $\angle KBC = \angle ABC$. Denote $S$ the intersection point of angle - bisectors of $\angle BKC$ and $\angle ACB$. Lines $AB$ and $KS$ intersect at point $L$, lines $BS$ and $CL$ intersect at point $M$ . Prove that line $KM$ passes through the midpoint of the segment $BC$.
Diagonals of an isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ are perpendicular. Let $DE$ be the perpendicular from $D$ to $AB$, and let $CF$ be the perpendicular from $C$ to $DE$. Prove that angle $DBF$ is equal to half of angle $FCD$.
Let $\triangle ABC$ such that $AB=AC$, $D$ and $E$ points on $AB$ and $BC$, respectively, with $DE\parallel AC$. Let $F$ on line $DE$ such that $CADF$ it´s a parallelogram. If $O$ is the circumcenter of $\triangle BDE$, prove that $O,F,A$ and $D$ lie on a circle.
Let $ABC$ be an isosceles triangle right at $C$ and $P$ any point on $CB$. Let also $Q$ be the midpoint of $AB$ and $R, S$ be the points on $AP$ such that $CR$ is perpendicular to $AP$ and $|AS|=|CR|$. Prove that the $|RS| = \sqrt2 |SQ|$.
Let $ABC$ be an isosceles acute triangle ($AB = BC$). On the side $BC$ we mark a point $P$, such that $\angle PAC = 45^o$, and $Q$ is the point of intersection of the perpendicular bisector of the segment $AP$ with the side $AB$. Prove that $PQ \perp BC$.
From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.
In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.
Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.
Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.
Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.
Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$
and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.
Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]