Problems

In the acute-angled triangle $ABC$, the heights $A{A}_1$ and $B{B}_1$ are drawn. Prove that $A_{1}C\times BC=B_{1}C\times AC$.

Let $A{A}_1$ and $B{B}_1$ be the heights of the triangle $ABC$. Prove that the triangles $A_1{B}_{1}C$ and $ABC$ are similar.

Let $G,F,H$ and $I$ be the midpoints of the sides $CD,DA,AB,BC$ of the square $ABCD$, whose area is equal to $S$. Find the area of the quadrilateral formed by the straight lines $BG,DH,AF,CE$.

a) Prove that if in the triangle the median coincides with the height then this triangle is an isosceles triangle.

b) Prove that if in a triangle the bisector coincides with the height then this triangle is an isosceles triangle.

Prove that the bisectors of a triangle intersect at one point.

A circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.