59th Ukrainian National Mathematical Olympiad

Question

Let **ABC** be a right triangle such that its legs satisfy the following: BC = 2AC. Show that the medians **AN** and **CM** are perpendicular to each other. (Danylo Khilko) FIGURE_0 **Fig. 6**

Accepted Answer

Since **AB** is a hypothenuse, then MCB = MBC (Fig. 6). On the other hand, AC^2 = CN CB, since BC = 2AC and CN = 12BC = 22AC. Then ACN ABC. Hence CAN = ABC. Thus, AKC = ACK + NAC = ACK + KCB = 90^, that finishes the proof.