Problems

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 $(a+b+c)/2$.

Two circles have radii $R_1$ and $R_2$, and the distance between their centers is $d$. Prove that these circles are orthogonal if and only if $d^2=R_1^2+R_2^2$.