HMMT 二月 2016 · 冲刺赛 · 第 20 题
HMMT February 2016 — Guts Round — Problem 20
题目详情
英文原题
- [ 11 ] Let ABC be a triangle with AB = 13, AC = 14, and BC = 15. Let G be the point on AC suchthat the reflection of BG over the angle bisector of ∠ B passes through the midpoint of AC . Let Y bethe midpoint of GC and X be a point on segment AG such that = 3. Construct F and H on ABAXand BC , respectively, such that F X ‖ BG ‖ HY . If AH and CF concur at Z and W is on AC such x gthat W Z ‖ BG , find W Z .
HMMT February 2016, February 20, 2016 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 11 ] Let ABC be a triangle with AB = 13, AC = 14, and BC = 15. Let G be the point on AC suchthat the reflection of BG over the angle bisector of ∠ B passes through the midpoint of AC . Let Y bethe midpoint of GC and X be a point on segment AG such that = 3. Construct F and H on ABAXand BC , respectively, such that F X ‖ BG ‖ HY . If AH and CF concur at Z and W is on AC such x gthat W Z ‖ BG , find W Z .
Proposed by: Ritesh Ragavender
√
1170 37
Answer:
1379
AG c 2
Observe that BG is the B -symmedian, and thus = . Stewart’s theorem gives us
GC a 2
√
√
2 2 2 2 2
√
2 a c b a b c ac 390 37
2 2 2
BG = − = 2( a + c ) − b = .
2 2 2 2 2 2
b ( a + c ) a + c a + c 197
Then by similar triangles,
√
ZA Y C ZA 1 6 1170 37
ZW = HY = BG = BG =
HA GC HA 2 7 1379
where is found with mass points or Ceva.ZAHA