In the acute-angled triangle $ABC$, with $AB \ne AC$, $D$ is the foot of the angle bisector of angle $A$, and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. The circumcircles of triangles $DBF$ and $DCE$ intersect for the second time at $M$. Prove that $ME = MF$. Leonard Giugiuc

In triangle $ABC$, points $M$, $N$ are the midpoints of sides $AB$, $AC$ respelctively. Let $D$ and $E$ be two points on line segment $BN$ such that $CD \parallel ME$ and $BD <BE$. Prove that $BD=2\cdot EN$.

On the sides $AD , BC$ of the square $ABCD$ the points $M, N$ are selected $N$, respectively, such that $AM = BN$. Point $X$ is the foot of the perpendicular from point $D$ on the line $AN$. Prove that the angle $MXC$ is right. (Mirchev Borislav)

Let $ABC$ be a triangle inscribed in circle $(I)$ that is tangent to the sides $BC,CA,AB$ at points $D,E, F$ respectively. Assume that $L$ is the intersection of $BE$ and $CF,G$ is the centroid of triangle $DEF,K$ is the symmetric point of $L$ about $G$. If $DK$ meets $EF$ at $P, Q$ is on $EF$ such that $QF = PE$, prove that $\angle DGE + \angle FGQ = 180^o$. Nguyễn Minh Hà

Let $ABCD$ be a parallelogram with $\angle BAD<90^o$ and $AB> BC$ . The angle bisector of $BAD$ intersects line $CD$ at point $P$ and line $BC$ at point $Q$. Prove that the center of the circle circumscirbed around the triangle $CPQ$ is equidistant from points $B$ and $D$.

Let $\vartriangle ABC$ be an equilateral triangle, and let $M$ and $N$ be points on $AB$ and $AC$, respectively, so that $AN = BM$ and $3MB = AB$. Lines $CM$ and $BN$ intersect at $O$. Find $\angle AOB$.

The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal.

Point $L$ is marked on the side $AB$ of a triangle $ABC$. The incircle of the triangle $ABC$ meets the segment $CL$ at points $P$ and $Q$ .Is it possible that the equalities $CP = PQ = QL$ hold if $CL$ is a) the median? b) the bisector? c) the altitude? d) the segment joining vertex $C$ with the point $L$ of tangency of the excircle of the triangie $ABC$ with $AB$ ? (I. Gorodnin)

Let $ABC$ be a triangle with $AC> AB$. Let $P$ be the intersection of $BC$ and the tangent through $A$ around the triangle $ABC$. Let $Q$ be the point on the straight line $AC$, so that $AQ = AB$ and $A$ is between $C$ and $Q$. Let $X$ and $Y$ be the center of $BQ$ and $AP$. Let $R$ be the point on $AP$ so that $AR = BP$ and $R$ is between $A$ and $P$. Show that $BR = 2XY$.

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.

Let $AB$ be a diameter of a circle $(\omega)$ with centre $O$. From an arbitrary point $M$ on $AB$ such that $MA < MB$ we draw the circles $(\omega_1)$ and $(\omega_2)$ with diameters $AM$ and $BM$ respectively. Let $CD$ be an exterior common tangent of $(\omega_1), (\omega_2)$ such that $C$ belongs to $(\omega_1)$ and $D$ belongs to $(\omega_2)$. The point $E$ is diametrically opposite to $C$ with respect to $(\omega_1)$ and the tangent to $(\omega_1)$ at the point $E$ intersects $(\omega_2)$ at the points $F, G$. If the line of the common chord of the circumcircles of the triangles $CED$ and $CFG$ intersects the circle $(\omega)$ at the points $K, L$ and the circle $(\omega_2)$ at the point $N$ (with $N$ closer to $L$), then prove that $KC = NL$.

Let $ABC$ be an acute triangle. Suppose the points $A',B',C'$ lie on its sides $BC,AC,AB$ respectively and the segments $AA',BB',CC'$ intersect in a common point $P$ inside the triangle. For each of those segments let us consider the circle such that the segment is its diameter, and the chord of this circle that contains the point $P$ and is perpendicular to this diameter. All three these chords occurred to have the same length. Prove that $P$ is the orthocenter of the triangle $ABC$. (Grigory Galperin)

Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.

The equal segments $AB$ and $CD$ intersect at the point $O$ and divide it by the relation $AO: OB = CO: OD = 1: 2 $. The lines $AD$ and $BC$ intersect at the point $M$. Prove that $DM = MB$.

An isosceles right-angled triangle $ABC$ is given. On the extensions of sides $AB$ and $AC$, behind vertices $B$ and $C$ equal segments $BK$ and $CL$ were laid. $E$ and F are the points of intersection of the segment $KL$ and the lines perpendicular to the $KC$ , passing through the points $B$ and $A$, respectively. Prove that $EF = FL$.

Consider $D$ the midpoint of the base $[BC]$ of the isosceles triangle ABC in which $\angle BAC < 90^o$. On the perpendicular from $B$ on the line $BC$ consider the point $E$ such that $\angle EAB= \angle BAC$, and on the line passing though $C$ parallel to the line $AB$ we consider the point $F$ such that $F$ and $D$ are on different side of the line $AC$ and $\angle FAC = \angle CAD$. Prove that $AE = CF$ and $BF = EF$

A semicircle is erected over the segment $AB$ with center $M$. Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$. The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$. Prove that $PM = PS$ holds. [i](Karl Czakler)[/i]

Given an $ABC$ triangle. Let $D$ be an extension of section $AB$ beyond $A$ such that that $AD = BC$ and $E$ is the extension of the section $BC$ beyond $B$ such that $BE = AC$. Prove that the circumcircle of triangle $DEB$ passes through the center of the inscribed circle of triangle $ABC$.

Let $ O $ be the center of the inscribed circle of the triangle $ ABC $, tangent to the side of $ BC $. Let $ M $ be the midpoint of $ AC $, and $ P $ be the intersection point of $ MO $ and $ BC $. Prove that $ AB = BP $ if $ \angle BAC = 2 \angle ACB $.

On the bisector of the angle $ BAC $ of the triangle $ ABC $ we choose the points $ {{B} _ {1}}, \, \, {{C} _ {1}} $ for which $ B {{B} _ {1 }}\perp AB $, $ C {{C} _ {1}} \perp AC $. The point $ M $ is the midpoint of the segment $ {{B} _ {1}} {{C} _ {1}} $. Prove that $ MB = MC $.

Let $\Gamma$ is a circle with diameter $AB$. Let $\ell$ be the tangent of $\Gamma$ at $A$, and $m$ be the tangent of $\Gamma$ through $B$. Let $C$ be a point on $\ell$, $C \ne A$, and let $q_1$ and $q_2$ be two lines that passes through $C$. If $q_i$ cuts $\Gamma$ at $D_i$ and $E_i$ ($D_i$ is located between $C$ and $E_i$) for $i = 1, 2$. The lines $AD_1, AD_2, AE_1, AE_2$ intersects $m$ at $M_1, M_2, N_1, N_2$ respectively. Prove that $M_1M_2 = N_1N_2$.

On side, $BC, AB$ of a parallelogram $ABCD$ lie points $M,N$ respectively such that $|AM| =|CN|$. Let $P$ be the intersection of $AM$ and $CN$. Prove that the angle bisector of $\angle APC$ passes through $D$.

Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal. [img]https://1.bp.blogspot.com/-tQEJ1VBCa8U/XzT7IhwlZHI/AAAAAAAAMVI/xpRdlj5Rl64VUt_tCRsQ1UxIsv_SGrMlACLcBGAsYHQ/s0/2014%2BMohr%2Bp3.png[/img]