The diagonals of a convex quadrilateral $ABCD$ intersect at $E$. The triangles $ABE, BCE, CDE$ and $DAE$ have centroids $K,L,M$ and $N$, and orthocentres $Q,R,S$ and $T$. Show that the quadrilaterals $QRST$ and $LMNK$ are similar.

Triangle $A'B'C'$ is located inside triangle $ABC$ such that $AB \parallel A'B' $, $BC \parallel B'C'$ and $CA \parallel C'A'$ , and all three sides of these parallel sides are at distance $d$ at each case. Let $O$ and $O'$ be the centers of the inscribed circles of the triangles $ABC$ and $A'B'C'$ and $K$ and $K'$ are the the centers of their circumcircles. Prove that points $O, O', K$ and $K'$ lie on a straight line.

The incentre of a triangle is joined by three segments to the three vertices of the triangle, thereby dividing it into three smaller triangles. If one of these three triangles is similar to the original triangle, find its angles. (A Shapovalov)

In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit?

Let $P$ be an arbitrary point in the plane of triangle $ABC$. Lines $PA, PB, PC$ meets the perpendicular bisectors of $BC,CA,AB$ at $O_a,O_b,O_c$ respectively. Let $(O_a)$ be the circle with center $O_a$ passing through two points $B,C$, two circles $(O_b), (O_c)$ are defined in the same manner. Two circles $(O_b), (O_c)$ meets at $A_1$, distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Let $Q$ be an arbitrary point in the plane of $ABC$ and $QB,QC$ meets $(O_c)$ and $(O_b)$ at $A_2,A_3$ distinct from $B,C$. Similarly, we have points $B_2,B_3,C_2,C_3$. Let $(K_a), (K_b), (K_c)$ be the circumcircles of triangles $A_1A_2A_3, B_1B_2B_3, C_1C_2C_3$. Prove that (a) three circles $(K_a), (K_b), (K_c)$ have a common point. (b) two triangles $K_aK_bK_c, ABC$ are similar. Trần Quang Hùng

The lengths of the sides of triangle $T'$ are equal to the lengths of the medians of triangle $T$. If triangles $T$ and $T'$ coincide in one angle, they are similar. Prove it.

We will say that a convex polygon $M$ has the property $(*)$ if the straight lines on which its sides lie, being moved outward by a distance of $1$ cm, form a polygon $M'$, similar to this one. a) Prove that if a convex polygon has property $(*)$ , then a circle can be inscribed in it. b) Find the fourth side of a quadrilateral satisfying condition $(*)$ if the lengths of its three consecutive sides are $9, 7$, and $3$ cm.

Determine whether there is a tetrahedron $ABCD$ with the longest edge of length 1 such that all its faces are similar right triangles with right angles at vertices $B,C.$ If so, determine which edge is the longest, which is the shortest and what is its length.

a) Let $S$ and $P$ be the area and perimeter of some triangle. The straight lines on which its sides are located move to the outside by a distance $h$. What will be the area and perimeter of the triangle formed by the three obtained lines? b) Let $V$ and $S$ be the volume and surface area of some tetrahedron. The planes on which its faces are located are moved to the outside by a distance $h$. What will be the volume and surface area of the tetrahedron formed by the three obtained planes?

Given a circle and a segment $ MN $. Find a point $ C $ on the circle such that the triangle $ ABC $, where $ A $ and $ B $ are the intersection points of the lines $ MC $ and $ NC $ with the circle, is similar to the triangle $ MNC $.

The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$. If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?

In a triangle $ABC$, points $P, Q$ and $ R$ distinct from the vertices of the triangle are chosen on sides $AB, BC$ and $CA$, respectively. The circumcircles of the triangles $APR$, $BPQ$, and $CQR$ are drawn. Prove that the centers of these circles are the vertices of a triangle similar to triangle $ABC$.

We assign to any quadrilateral $ ABCD $ the centers of the circles circumscribed in the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $. Prove that if the vertices of a convex quadrilateral $ Q $ do not lie on the circle, then a) the four points assigned to quadrilateral Q in the above manner are the vertices of the convex quadrilateral. Let us denote this quadrilateral by $ t(Q) $, b) the vertices of the quadrilateral $ t(Q) $ do not lie on the circle, c) quadrilaterals $ Q $ and $ t(t(Q) $ are similar.

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$, two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$. Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$.

(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles. (b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.

The circles $k_1$ and $k_2$ are tangent at the point $P$. A line is drawn through $P$, cutting $k_1$ at $A_1$ and $k_2$ at $A_2$. A second line is drawn through $P$, cutting $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that the triangles $PA_1B_1$ and $PA_2B_2$ are similar.