HMMT 二月 2023 · 几何 · 第 1 题

HMMT February 2023 — Geometry — Problem 1

Let ABCDEF be a regular hexagon, and let P be a point inside quadrilateral ABCD . If the area of triangle P BC is 20, and the area of triangle P AD is 23, compute the area of hexagon ABCDEF .

解析

Let ABCDEF be a regular hexagon, and let P be a point inside quadrilateral ABCD . If the area of triangle P BC is 20, and the area of triangle P AD is 23, compute the area of hexagon ABCDEF . Proposed by: Ankit Bisain Answer: 189 Solution: If s is the side length of the hexagon, h is the length of the height from P to BC , and h 1 2 1 1 is the length of the height from P to AD , we have [ P BC ] = s · h and [ P AD ] = (2 s ) · h . We also 1 2 2 2 √ 3 have h + h = s . Therefore, 1 2 2 √ 3 2 2[ P BC ] + [ P AD ] = s ( h + h ) = s . 1 2 2 √ 3 3 2 The area of a hexagon with side length s is s , giving a final answer of 2 6[ P BC ] + 3[ P AD ] = 6 · 20 + 3 · 23 = 189 .