From an arbitrary point \(M\) lying within a given angle with vertex \(A\), the perpendiculars \(MP\) and \(MQ\) are dropped to the sides of the angle. From point \(A\), the perpendicular \(AK\) is dropped to the segment \(PQ\). Prove that \(\angle PAK = \angle MAQ\).
On a circle, the points \(A, B, C, D\) are given in the indicated order. \(M\) is the midpoint of the arc \(AB\). We denote the intersection points of the chords \(MC\) and \(MD\) with the chord \(AB\) by \(E\) and \(K\). Prove that \(KECD\) is an inscribed quadrilateral.
Two circles intersect at the points \(P\) and \(Q\). Through the point \(A\) of the first circle, the lines \(AP\) and \(AQ\) are drawn intersecting the second circle at points \(B\) and \(C\). Prove that the tangent at point \(A\) to the first circle is parallel to the line \(BC\).
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 \(M.\)
From an arbitrary point \(M\) on the side \(BC\) of the right angled triangle \(ABC\), the perpendicular \(MN\) is dropped onto the hypotenuse \(AB\). Prove that \(\angle MAN = \angle MCN\).
The diagonals of the trapezium \(ABCD\) with the bases \(AD\) and \(BC\) intersect at the point \(O\); the points \(B'\) and \(C'\) are symmetrical to the vertices \(B\) and \(C\) with respect to the bisector of the angle \(BOC\). Prove that \(\angle C'AC = \angle B'DB\).
On the circle, the points \(A, B, C\) and \(D\) are given. The lines \(AB\) and \(CD\) intersect at the point \(M\). Prove that \(AC \times AD / AM = BC \times BD / BM\).
In the triangle \(ABC\), the height \(AH\) is drawn; \(O\) is the center of the circumscribed circle. Prove that \(\angle OAH = | \angle B - \angle C\)|.
Prove that from the point \(C\) lying outside of the circle we can draw exactly two tangents to the circle and the lengths of these tangents (that is, the distance from \(C\) to the points of tangency) are equal.