HMMT 二月 2012 · 冲刺赛 · 第 9 题
HMMT February 2012 — Guts Round — Problem 9
题目详情
英文原题
[5] Given △ABC with AB < AC , the altitude AD, angle bisector AE, and median AFare drawn fromA, with D, E, F all lying on BC. If∡BAD = 2∡DAE = 2∡EAF =∡FAC , what are all possible values of ∡ACB ?
解析
英文解析
- [ 5 ] Given △ ABC with AB < AC , the altitude AD , angle bisector AE , and median AF are drawnfrom A , with D, E, F all lying on BC . If ∡ BAD = 2 ∡ DAE = 2 ∡ EAF = ∡ F AC , what are all possiblevalues 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 .
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