In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

Is there a convex pentagon in which each diagonal is equal to a side?

In triangle $ABC$, the perpendicular bisectors of sides $AB$ and $BC$ intersect side $AC$ at points $P$ and $Q$, respectively, with point $P$ lying on the segment $AQ$. Prove that the circumscribed circles of the triangles $PBC$ and $QBA$ intersect on the bisector of the angle $PBQ$.

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

Inside an isosceles right triangle $ABC$ with hypotenuse $AB$ a point $M$ is taken such that the angle $\angle MAB$ is $15 ^o$ larger than the angle $\angle MAC$ , and the angle $\angle MCB$ is $15^o$ larger than the angle $\angle MBC$. Find the angle $\angle BMC$ .

In trapezoid $ABCD$: $BC <AD, AB = CD, K$ is midpoint of $AD, M$ is midpoint of $CD, CH$ is height. Prove that lines $AM, CK$ and $BH$ intersect at one point.

Is it possible to cut a regular triangular prism into two equal pyramids?

The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.

The medians $AA_0, BB_0$, and $CC_0$ of the acute-angled triangle $ABC$ intersect at the point $M$, and heights $AA_1, BB_1$ and $CC_1$ at point $H$. Tangent to the circumscribed circle of triangle $A_1B_1C_1$ at $C_1$ intersects the line $A_0B_0$ at the point $C'$. Points $A'$ and $B'$ are defined similarly. Prove that $A', B'$ and $C'$ lie on one line perpendicular to the line $MH$.

Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.