Given a triangle ABC, inside which the point M is marked. On the sides BC,CA and AB the following points A1,B1 and C1 are chosen, respectively, that MA1∥CA, MB1∥AB, MC1∥BC. Let S be the area of triangle ABC,QM be the area of the triangle A1B1C1.
a) Prove that if the triangle ABC is acute, and M is the point of intersection of its altitudes , then 3QM≤S. Is there such a number k>0 that for any acute-angled triangle ABC and the point M of intersection of its altitudes, such thatthe inequality QM>kS holds?
b) For cases where the point M is the point of intersection of the medians, the center of the inscribed circle, the center of the circumcircle, find the largest k1>0 and the smallest k2>0 such that for an arbitrary triangle ABC, holds the inequality k1S≤QM≤k2S (for the center of the circumscribed circle, only acute-angled triangles ABC are considered). geometrygeometric inequalityUkrainian TYM