HMMT 二月 2015 · 几何 · 第 9 题

HMMT February 2015 — Geometry — Problem 9

Let ABCD be a regular tetrahedron with side length 1. Let X be the point in triangle BCD such that [ XBC ] = 2[ XBD ] = 4[ XCD ], where [ ̟ ] denotes the area of figure ̟ . Let Y lie on segment AX such that 2 AY = Y X . Let M be the midpoint of BD . Let Z be a point on segment AM such that AZ the lines Y Z and BC intersect at some point. Find . ZM

解析

Let ABCD be a regular tetrahedron with side length 1. Let X be the point in triangle BCD such that [ XBC ] = 2[ XBD ] = 4[ XCD ], where [ ̟ ] denotes the area of figure ̟ . Let Y lie on segment AX such that 2 AY = Y X . Let M be the midpoint of BD . Let Z be a point on segment AM such that AZ the lines Y Z and BC intersect at some point. Find . ZM 4 Answer: We apply three-dimensional barycentric coordinates with reference tetrahedron 7 ABCD . The given conditions imply that X = (0 : 1 : 2 : 4) Y = (14 : 1 : 2 : 4) M = (0 : 1 : 0 : 1) Z = ( t : 1 : 0 : 1) ( ) ( ) 14 1 2 4 t 1 1 for some real number t . Normalizing, we obtain Y = , , , and Z = , , 0 , . If 21 21 21 21 t +2 t +2 t +2 Y Z intersects line BC then there exist parameters α + β = 1 such that αY + βZ has zero A and D coordinates, meaning 14 t α + β = 0 21 t + 2 4 1 α + β = 0 21 t + 2 α + β = 1 . 22 7 Adding twice the second equation to the first gives α + β = 0, so α = − 22, β = 21, and thus t = . 21 2 AZ 2+2 4 It follows that Z = (7 : 2 : 0 : 2), and = = . ZM 7 7