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HMMT 二月 2011 · ALGGEO 赛 · 第 12 题

HMMT February 2011 — ALGGEO Round — Problem 12

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let ABCDEF be a convex equilateral hexagon such that lines BC , AD , and EF are parallel. Let H be the orthocenter of triangle ABD . If the smallest interior angle of the hexagon is 4 degrees, determine the smallest angle of the triangle HAD in degrees.
解析
  1. Let ABCDEF be a convex equilateral hexagon such that lines BC , AD , and EF are parallel. Let H be the orthocenter of triangle ABD . If the smallest interior angle of the hexagon is 4 degrees, determine the smallest angle of the triangle HAD in degrees. Answer: 3 Algebra & Geometry Individual Test A ′ A F B H ′ B E C ′ D D A F B H E C D Note that ABCD and DEF A are isosceles trapezoids, so ∠ BAD = ∠ CDA and ∠ F AD = ∠ EDA . In ◦ order for the hexagon to be convex, the angles at B , C , E , and F have to be obtuse, so ∠ A = ∠ D = 4 . Letting s be a side length of the hexagon, AD = AB cos ∠ BAD + BC + CD cos ∠ CDA = s (1 + 2 cos ∠ BAD ), so ∠ BAD is uniquely determined by AD . Since the same equation holds for trapezoid ◦ ◦ ◦ ◦ DEF A , it follows that ∠ BAD = ∠ F AD = ∠ CDA = ∠ EDA = 2 . Then ∠ BCD = 180 − 2 = 178 . ◦ ◦ ◦ Since 4 BCD is isosceles, ∠ CDB = 1 and ∠ BDA = 1 . (One may also note that ∠ BDA = 1 by observing that equal lengths AB and BC must intercept equal arcs on the circumcircle of isosceles trapezoid ABCD ). ′ ′ ′ Let A , B , and D be the feet of the perpendiculars from A , B , and D to BD , DA , and AB , respectively. Angle chasing yields ′ ′ ◦ ′ ′ ◦ ′ ′ ∠ AHD = ∠ AHB + ∠ DHB = (90 − ∠ A AB ) + (90 − ∠ D DB ) ◦ ◦ ◦ = ∠ BDA + ∠ BAD = 1 + 2 = 3 ◦ ′ ◦ ∠ HAD = 90 − ∠ AHB = 89 ◦ ′ ◦ ∠ HDA = 90 − ∠ DHB = 88 ◦ Hence the smallest angle in 4 HAD is 3 . It is faster, however, to draw the circumcircle of DEF A , and to note that since H is the orthocenter of triangle ABD , B is the orthocenter of triangle HAD . Then since F is the reflection of B across ◦ ◦ ◦ AD , quadrilateral HAF D is cyclic, so ∠ AHD = ∠ ADF + ∠ DAF = 1 + 2 = 3 , as desired. Algebra & Geometry Individual Test