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 \(ABC\).
Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.
In the acute-angled triangle \(ABC\), the heights \(AA_1\) and \(BB_1\) are drawn. Prove that \(A_1C \times BC = B_1C \times AC\).
Let \(AA_1\) and \(BB_1\) be the heights of the triangle \(ABC\). Prove that the triangles \(A_1B_1C\) and \(ABC\) are similar.
A unit square is divided into \(n\) triangles. Prove that one of the triangles can be used to completely cover a square with side length \(\frac{1}{n}\).