Problems

The circles ${\sigma }_1$ and ${\sigma }_2$ intersect at points $A$ and $B$. At the point $A$ to ${\sigma }_1$ and ${\sigma }_2$, respectively, the tangents $l_1$ and $l_2$ are drawn. The points $T_1$ and $T_2$ are chosen respectively on the circles ${\sigma }_1$ and ${\sigma }_2$ so that the angular measures of the arcs $T_{1}A$ and $A{T}_2$ are equal (the arc value of the circle is considered in the clockwise direction). The tangent $t_1$ at the point $T_1$ to the circle ${\sigma }_1$ intersects $l_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle ${\sigma }_2$ intersects $l_1$ at the point $M_2$. Prove that the midpoints of the segments $M_1{M}_2$ are on the same line, independent of the positions of the points $T_1,T_2$.

Two circles of radius $R$ touch at point $E$. On one of them, point $B$ is chosen and on the other point $D$ is chosen. These points have a property of $\mathrm{\angle }BED={90}^{\circ }$. Prove that $BD=2R$.

Two circles of radius $R$ intersect at points $D$ and $B$. Let $F$ and $G$ be the points of intersection of the middle perpendicular to the segment $BD$ with these circles lying on one side of the line $BD$. Prove that $B{D}^2+F{G}^2=4R^2$.

Prove that a convex $n$-gon is regular if and only if it is transformed into itself when it is rotated through an angle of ${360}^{\circ }/n$ with respect to some point.

Two perpendicular straight lines are drawn through the centre of the square. Prove that their intersection points with the sides of a square form a square.

Two circles $c$ and $d$ are tangent at point $B$. Two straight lines intersecting the first circle at points $D$ and $E$ and the second circle at points $G$ and $F$ are drawn through the point $B$. Prove that $ED$ is parallel to $FG$.

Several circles, whose total length of circumferences is 10, are placed inside a square of side 1. Prove that there will always be some straight line that crosses at least four of the circles.