HMMT 十一月 2011 · 冲刺赛 · 第 22 题

HMMT November 2011 — Guts Round — Problem 22

[ 12 ] Let ABC be a triangle with AB = 23, BC = 24, and CA = 27. Let D be the point on segment AC such that the incircles of triangles BAD and BCD are tangent. Determine the ratio CD/DA .

解析

[ 12 ] Let ABC be a triangle with AB = 23, BC = 24, and CA = 27. Let D be the point on segment AC such that the incircles of triangles BAD and BCD are tangent. Determine the ratio CD/DA . 14 Answer: Let X, Z, E be the points of tangency of the incircle of ABD to AB, BD, DA respec- 13 tively. Let Y, Z, F be the points of tangency of the incircle of CBD to CB, BD, DC respectively. We note that CB + BD + DC = CY + Y B + BZ + ZD + DF + F C = 2( CY ) + 2( BY ) + 2( DF )2(24) + 2( DF ) by equal tangents, and that similarly AB + BD + DA = 2(23) + 2( DE ) . Since DE = DZ = DF by equal tangents, we can subtract the equations above to get that CB + CD − AB − AD = 2(24) − 24(23) ⇒ CD − DA = 1 . 14 Since we know that CD + DA = 27, we get that CD = 14, DA = 13, so the desired ratio is . 13 Guts Round B Y X Z A E D F C