A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.

In a certain triangle, the bisectors of the two interior angles were extended to the intersection with the circumscribed circle and two equal chords were obtained. Is it true that the triangle is isosceles?

In the regular hexagon $ABCDEF$ on the line $AF$, the point $X$ is taken so that the angle $XCD$ is $45^o$. Find the angle $\angle FXE$. (Kiev Olympiad)

A circle can be circumscribed around the quadrilateral $ABCD$. Point $P$ is the foot of the perpendicular drawn from point $A$ on line $BC$, and respectively $Q$ from $A$ on $DC$, $R$ from $D$ on $AB$ and $T$ from $D$ on $BC$ . Prove that points $P,Q,R$ and $T$ lie on the same circle. (A. Myakishev)

Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides. (A. Zaslavsky)

Opposite sides of a convex hexagon are parallel. Let's call the "height" of such a hexagon a segment with ends on straight lines containing opposite sides and perpendicular to them. Prove that a circle can be circumscribed around this hexagon if and only if its "heights" can be parallelly moved so that they form a triangle. (A. Zaslavsky)

Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar? (D. Shnol)

The radii $r$ and $R$ of two non-intersecting circles are given. The common internal tangents of these circles are perpendicular. Find the area of the triangle bounded by these tangents, as well as the common external tangents.

Given a quadrilateral $ABCD$. $A ', B', C'$ and $D'$ are the midpoints of the sides $BC, CB, BA$ and $AB$, respectively. It is known that $AA'= CC'$, $BB'= DD'$. Is it true that $ABCD$ is a parallelogram? (M. Volchkevich)

Angle $A$ in triangle $ABC$ is equal to $120^o$. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to $AB + AC$. (V. Protasov)

There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)? (S. Markelov).

Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$. (A. Zaslavsky)