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 $$\overrightarrow{AM}+\overrightarrow{BK}+\overrightarrow{CP}=0$$ 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 $\beta =3\alpha$ (Fig.).

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