Problems

Two circles touch at a point $A$. A common (outer) tangent touching the circles at points $C$ and $B$ is drawn. Prove that $\mathrm{\angle }CAB={90}^{\circ }$.

Two circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ touch at the point $A$. A straight line intersects $S_1$ at $A_1$ and $S_2$ at the point $A_2$. Prove that $O_1{A}_1\parallel O_2{A}_2$.

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 $S_{ABE}+S_{CDE}=S_{BCE}+S_{ADE}$.

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 $BG,DH,AF,CE$.

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

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.