HMMT 二月 2005 · 几何 · 第 3 题

HMMT February 2005 — Geometry — Problem 3

Let ABCD be a rectangle with area 1, and let E lie on side CD . What is the area of the triangle formed by the centroids of triangles ABE , BCE , and ADE ? ◦ ◦ 6 6

解析

Let ABCD be a rectangle with area 1, and let E lie on side CD . What is the area of the triangle formed by the centroids of triangles ABE , BCE , and ADE ? Solution: 1/9 Let the centroids of ABE , BCE , and ADE be denoted by X , Y , and Z , respectively. Let d ( P, QR ) denote the distance from P to line QR . Since the centroid lies two-thirds of the distance from each vertex to the midpoint of the opposite edge, d ( X, AB ) = 1 d ( Y, CD ) = d ( Z, CD ) = BC , so Y Z is parallel to CD and d ( X, Y Z ) = BC − 3 2 1 1 1 BC = BC . Likewise, d ( Z, AD ) = DE and d ( Y, BC ) = CE , so that since Y Z 3 3 3 3 1 2 is perpendicular to AD and BC , we have that Y Z = CD − ( DE + CE ) = CD . 3 3 1 1 2 1 1 Therefore, the area of XY Z is ( BC )( CD ) = BC · CD = . 2 3 3 9 9 A B C D E ◦ ◦ 6 6