HMMT 二月 2012 · 几何 · 第 3 题

HMMT February 2012 — Geometry — Problem 3

Let ABC be a triangle with incenter I . Let the circle centered at B and passing through I intersect side AB at D and let the circle centered at C passing through I intersect side AC at E . Suppose DE is the perpendicular bisector of AI . What are all possible measures of angle BAC in degrees?

解析

Let ABC be a triangle with incenter I . Let the circle centered at B and passing through I intersect side AB at D and let the circle centered at C passing through I intersect side AC at E . Suppose DE is the perpendicular bisector of AI . What are all possible measures of angle BAC in degrees? 540 Answer: Let α = ∡ BAC . DE is the perpendicular bisector of AI , so DA = DI , and 7 ◦ ∠ DIA = ∠ DAI = α/ 2. Thus, ∠ IDB = ∠ DIB = α , since BD = BI . This gives ∠ DBI = 180 − 2 α , ◦ ◦ so that ∠ ABC = 360 − 4 α . Similarly, ∠ ACB = 360 − 4 α . Now, summing the angles in ABC , we ◦ 540 ◦ ◦ find 720 − 7 α = 180 , so that α = . 7