The isosceles trapeziums \(ABCD\) and \(A_1B_1C_1D_1\) with corresponding parallel sides are inscribed in a circle. Prove that \(AC = A_1C_1\).
From the point \(M\), moving along a circle the perpendiculars \(MP\) and \(MQ\) are dropped onto the diameters \(AB\) and \(CD\). Prove that the length of the segment \(PQ\) does not depend on the position of the point \(M.\)
Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than \(k\) chords, then the total length of all the chords is less than \(\pi k\).