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3(A_1B_1C_1) <= S , max k_1, mion k_2 so that k_1 S<=(A_1B_1C_1) <= k_2 S

Source: XII All-Ukrainian Tournament of Young Mathematicians, Qualifying p17

May 27, 2021
geometrygeometric inequalityUkrainian TYM

Problem Statement

Given a triangle ABCABC, inside which the point MM is marked. On the sides BC,CABC,CA and ABAB the following points A1,B1A_1,B_1 and C1C_1 are chosen, respectively, that MA1∥CAMA_1 \parallel CA, MB1∥ABMB_1 \parallel AB, MC1∥BCMC_1 \parallel BC. Let S be the area of ​​triangle ABC,QMABC, Q_M be the area of ​​the triangle A1B1C1A_1 B_1 C_1. a) Prove that if the triangle ABCABC is acute, and M is the point of intersection of its altitudes , then 3QM≤S3Q_M \le S. Is there such a number k>0k> 0 that for any acute-angled triangle ABCABC and the point MM of intersection of its altitudes, such thatthe inequality QM>kSQ_M> k S holds? b) For cases where the point MM is the point of intersection of the medians, the center of the inscribed circle, the center of the circumcircle, find the largest k1>0k_1> 0 and the smallest k2>0k_2> 0 such that for an arbitrary triangle ABCABC, holds the inequality k1S≤QM≤k2Sk_1S \le Q_M\le k_2S (for the center of the circumscribed circle, only acute-angled triangles ABCABC are considered).
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