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.
Prove that the triangle \(ABC\) is regular if and only if, by turning it by \(60^{\circ}\) (either clockwise or anticlockwise) with respect to point A, its vertex B moves to \(C\).
Prove that the midpoints of the sides of a regular polygon form a regular polygon.
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.
Prove that if you rotate through an angle of \(\alpha\) with the center at the origin, the point with the coordinates \((x, y)\), it goes to the point \((x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha)\).
Prove that under homothety, a circle transforms into a circle.
Two circles touch at point \(K\). The line passing through point \(K\) intersects these circles at points \(A\) and \(B\). Prove that the tangents to the circles drawn through points \(A\) and \(B\) are parallel.
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 \(FG\).
Prove that the points symmetric to an arbitrary point relative to the midpoints of the sides of a square are vertices of some square.
The points \(A\) and \(B\) and the line \(l\) are given on a plane. On which trajectory does the intersection point of the medians of the triangles \(ABC\) move, if the point \(C\) moves along the line \(l\)?