HMMT 十一月 2014 · 冲刺赛 · 第 17 题

HMMT November 2014 — Guts Round — Problem 17

[ 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 circle is called an A -mixtilinear excircle ). Find the radius of Ω.

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 circle is called an A -mixtilinear excircle ). Find the radius of Ω. 75 Answer: Let M be the midpoint of BC . Let D be the point diametrically opposite A on the 8 circumcircle, and let the A -mixtilinear excircle be tangent to lines AB and AC at X and Y . Let O be the center of the A -mixtilinear excircle. Notice that △ AOX ∼ △ ABM . If we let x be the desired radius, we have x + AD 5 = . x 3 AD 5 25 We can compute = since △ ADB ∼ △ ABM , we derive AD = . From here it follows that 5 4 4 75 x = . 8 Guts Round