Problems

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

From a point $A$ the tangents $AB$ and $AC$ are drawn to a circle with center $O$. Prove that if from the point $M$ the segment $AO$ is visible at an angle of ${90}^{\circ }$, then the segments $OB$ and $OC$ are also visible from it at equal angles.

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 $d^2=R_1^2+R_2^2$.

Let $E$ and $F$ be the midpoints of the sides $BC$ and $AD$ of the parallelogram $ABCD$. Find the area of the quadrilateral formed by the lines $AE,ED,BF$ and $FC$, if it is known that the area $ABCD$ is equal to $S$.

A polygon is drawn around a circle of radius $r$. Prove that its area is equal to $pr$, where $p$ is the semiperimeter of the polygon.

The point $E$ is located inside the parallelogram $ABCD$. Prove that $S_{ABE}+S_{CDE}=S_{BCE}+S_{ADE}$.

The diagonals of the quadrilateral $ABCD$ intersect at the point $O$. Prove that $S_{AOB}=S_{COD}$ if and only if $BC\parallel AD$.

Prove that a convex quadrilateral $ABCD$ can be drawn inside a circle if and only if $\mathrm{\angle }ABC+\mathrm{\angle }CDA={180}^{\circ }$.

Prove that a convex quadrilateral $ICEF$ can contain a circle if and only if $IC+EH=CE+IF$.