Problems

On the sides \(AB\), \(BC\) and \(AC\) of the triangle \(ABC\) points \(P\), \(M\) and \(K\) are chosen so that the segments \(AM\), \(BK\) and \(CP\) intersect at one point and \[\vec{AM} + \vec{BK}+\vec{CP} = 0\] Prove that \(P\), \(M\) and \(K\) are the midpoints of the sides of the triangle \(ABC\).

Prove that the area \(S_{ABC}\) of a triangle is equal to \(abc/4R\).

The point \(D\) lies on the base \(AC\) of the isosceles triangle \(ABC\). Prove that the radii of the circumscribed circles of the triangles \(ABD\) and \(CBD\) are equal.

Express the area of the triangle \(ABC\) through the length of the side \(BC\) and the angles \(B\) and \(C\).

Two intersecting circles of radius \(R\) are given, and the distance between their centers is greater than \(R\). Prove that \(\angle ECD = 3\angle CAD\).

Find all triangles in which the angles form an arithmetic progression, and the sides form: a) an arithmetic progression; b) a geometric progression.