HMMT 二月 2012 · 冲刺赛 · 第 9 题

HMMT February 2012 — Guts Round — Problem 9

[ 5 ] Given △ ABC with AB < AC , the altitude AD , angle bisector AE , and median AF are drawn from A , with D, E, F all lying on BC . If ∡ BAD = 2 ∡ DAE = 2 ∡ EAF = ∡ F AC , what are all possible values of ∡ ACB ? 2

解析

[ 5 ] Given △ ABC with AB < AC , the altitude AD , angle bisector AE , and median AF are drawn from A , with D, E, F all lying on BC . If ∡ BAD = 2 ∡ DAE = 2 ∡ EAF = ∡ F AC , what are all possible values of ∡ ACB ? ◦ Answer: 30 or π/ 6 radians Let H and O be the orthocenter and circumcenter of ABC , respctively: it is well-known (and not difficult to check) that ∡ BAH = ∡ CAO . However, note that ∡ BAH = ∡ BAD = ∡ CAF , so ∡ CAF = ∡ CAO , that is, O lies on median AF , and since AB < AC , it follows ◦ that F = O . Therefore, ∡ BAC = 90 . 2 ◦ Now, we compute ∡ ACB = ∡ BAD = ∡ BAC = 30 . 6 2