Problems

The bisector of the outer corner at the vertex $C$ of the triangle $ABC$ intersects the circumscribed circle at the point $D$. Prove that $AD=BD$.

The vertex $A$ of the acute-angled triangle $ABC$ is connected by a segment with the center $O$ of the circumscribed circle. The height $AH$ is drawn from the vertex $A$. Prove that $\mathrm{\angle }BAH=\mathrm{\angle }OAC$.

The vertex $A$ of the acute-angled triangle $ABC$ is connected by a segment with the center $O$ of the circumscribed circle. The height $AH$ is drawn from the vertex $A$. Prove that $\mathrm{\angle }BAH=\mathrm{\angle }OAC$.

From an arbitrary point $M$ lying within a given angle with vertex $A$, the perpendiculars $MP$ and $MQ$ are dropped to the sides of the angle. From point $A$, the perpendicular $AK$ is dropped to the segment $PQ$. Prove that $\mathrm{\angle }PAK=\mathrm{\angle }MAQ$.

Prove that from the point $C$ lying outside of the circle we can draw exactly two tangents to the circle and the lengths of these tangents (that is, the distance from $C$ to the points of tangency) are equal.

Two circles intersect at points $A$ and $B$. Point $X$ lies on the line $AB$, but not on the segment $AB$. Prove that the lengths of all of the tangents drawn from $X$ to the circles are equal.

Let $a$ and $b$ be the lengths of the sides of a right-angled triangle and $c$ the length of its hypotenuse. Prove that:

a) The radius of the inscribed circle of the triangle is $(a+b-c)/2$;

b) The radius of the circle that is tangent to the hypotenuse and the extensions of the sides of the triangle, is equal to $(a+b+c)/2$.

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.