HMMT 二月 2007 · 几何 · 第 8 题

HMMT February 2007 — Geometry — Problem 8

[ 6 ] ABCD is a convex quadrilateral such that AB < AD . The diagonal AC bisects ∠ BAD , and ◦ ◦ m ∠ ABD = 130 . Let E be a point on the interior of AD , and m ∠ BAD = 40 . Given that BC = CD = DE , determine m ∠ ACE in degrees.

解析

[ 6 ] ABCD is a convex quadrilateral such that AB < AD . The diagonal AC bisects ∠ BAD , and ◦ ◦ m ∠ ABD = 130 . Let E be a point on the interior of AD , and m ∠ BAD = 40 . Given that BC = CD = DE , determine m ∠ ACE in degrees. 2 ◦ ′ Answer: 55 . First, we check that ABCD is cyclic. Reflect B over AC to B on AD , and note that ′ ′ ′ ◦ ′ ◦ B C = CD . Therefore, m ∠ ADC = m ∠ B DC = m ∠ CB D = 180 − m ∠ AB C = 180 − m ∠ CBA . ◦ ◦ ◦ Now m ∠ CBD = m ∠ CAD = 20 and m ∠ ADC = 180 − m ∠ CBA = 30 . Triangle CDE is isosceles, ◦ ◦ ◦ ◦ so m ∠ CED = 75 and m ∠ AEC = 105 . It follows that m ∠ ECA = 180 − m ∠ AEC − m ∠ CAE = 55 .