HMMT 二月 2013 · 几何 · 第 8 题
HMMT February 2013 — Geometry — Problem 8
题目详情
英文原题
Let ABCD be a convex quadrilateral. Extend line CD pastDto meet line ABatPand extend line
CB pastBto meet line AD atQ. Suppose that line ACbisects ∠BAD . IfAD =7
4,AP=21
2, and
AB=14
11, compute AQ.
解析
英文解析
- Let ABCD be a convex quadrilateral. Extend line CD past D to meet line AB at P and extend line
7 21
CB past B to meet line AD at Q . Suppose that line AC bisects ∠ BAD . If AD = , AP = , and
4 2
AB = , compute AQ .14
42 1 1 1 111
Answer: We prove the more general statement + = + , from which the answer
13 AB AP AD AQeasily follows.
Denote ∠ BAC = ∠ CAD = γ , ∠ BCA = α , ∠ ACD = β . Then we have that by the law of sines,
sin( γ + α ) sin( γ − β ) sin( γ − α ) sin( γ + β )
AC AC AC AC + = + = + = + where we have simply used the sine
AB AP sin( α ) sin( β ) sin( α ) sin( β ) AD AQaddition formula for the middle step.
Dividing the whole equation by AC gives the desired formula, from which we compute AQ = ( +11
2 4 − 1 4214 − ) = .
21 7 13