HMMT 十一月 2014 · 冲刺赛 · 第 17 题
HMMT November 2014 — Guts Round — Problem 17
题目详情
英文原题
- [ 10 ] Let ABC be a triangle with AB = AC = 5 and BC = 6. Denote by ω the circumcircle of ABC .
We draw a circle Ω which is externally tangent to ω as well as to the lines AB and AC (such a circleis called an A -mixtilinear excircle ). Find the radius of Ω.
解析
英文解析
- [ 10 ] Let ABC be a triangle with AB = AC = 5 and BC = 6. Denote by ω the circumcircle of ABC .
We draw a circle Ω which is externally tangent to ω as well as to the lines AB and AC (such a circleis called an A -mixtilinear excircle ). Find the radius of Ω.
Answer: Let M be the midpoint of BC . Let D be the point diametrically opposite A on the 75
circumcircle, and let the A -mixtilinear excircle be tangent to lines AB and AC at X and Y . Let O be 8
the center of the A -mixtilinear excircle.
Notice that △ AOX ∼ △ ABM . If we let x be the desired radius, we havex + AD 5 = .
x 3
AD 5 25
We can compute = since △ ADB ∼ △ ABM , we derive AD = . From here it follows that
5 4 4
x = .75
Guts Round 8