HMMT 十一月 2012 · 团队赛 · 第 9 题

HMMT November 2012 — Team Round — Problem 9

[ 5 ] Triangle ABC satisfies ∠ B > ∠ C . Let M be the midpoint of BC , and let the perpendicular bisector of BC meet the circumcircle of △ ABC at a point D such that points A , D , C , and B appear ◦ ◦ on the circle in that order. Given that ∠ ADM = 68 and ∠ DAC = 64 , find ∠ B . ′ ′ ′ ′ ′

解析

[ 5 ] Triangle ABC satisfies ∠ B > ∠ C . Let M be the midpoint of BC , and let the perpendicular bisector of BC meet the circumcircle of △ ABC at a point D such that points A , D , C , and B appear ◦ ◦ on the circle in that order. Given that ∠ ADM = 68 and ∠ DAC = 64 , find ∠ B . ◦ Answer: 86 Extend DM to hit the circumcircle at E . Then, note that since ADEB is a cyclic ◦ ◦ ◦ ◦ ◦ quadrilateral, ∠ ABE = 180 − ∠ ADE = 180 − ∠ ADM = 180 − 68 = 112 . ◦ We also have that ∠ M EC = ∠ DEC = ∠ DAC = 64 . But now, since M is the midpoint of BC ◦ and since EM ⊥ BC , triangle BEC is isosceles. This implies that ∠ BEM = ∠ M EC = 64 , and ◦ ◦ ◦ ◦ ◦ ∠ M BE = 90 − ∠ M EB = 26 . It follows that ∠ B = ∠ ABE − ∠ M BE = 112 − 26 = 86 . ′ ′ ′ ′ ′