Problems

Prove that, when a circle is translated it becomes a circle.

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 $B{D}^2+F{G}^2=4R^2$.

Inside the rectangle $ABCD$, the point $E$ is taken. Prove that there exists a convex quadrilateral with perpendicular diagonals of lengths $AB$ and $BC$ whose sides are equal to $AE$, $BE$, $CE$, $DE$.