Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).
Two intersecting circles of radius \(R\) are given, and the distance between their centers is greater than \(R\). Prove that \(\beta = 3\alpha\) (Fig.).
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 \(\angle BED = 90^{\circ}\). 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 \(BD^2 + FG^2 = 4R^2\).
A parallelogram \(ABCD\) and a point \(E\) are given. Through the points \(A, B, C, D\), lines parallel to the straight lines \(EC, ED, EA,EB\), respectively, are drawn. Prove that they intersect at one point.
Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.
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 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.
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 \(FG\).