HMMT 二月 2013 · 几何 · 第 8 题

HMMT February 2013 — Geometry — Problem 8

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 14 AB = , compute AQ . 11

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 14 AB = , compute AQ . 11 42 1 1 1 1 Answer: We prove the more general statement + = + , from which the answer 13 AB AP AD AQ easily 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 AQ addition formula for the middle step. 11 Dividing the whole equation by AC gives the desired formula, from which we compute AQ = ( + 14 2 4 − 1 42 − ) = . 21 7 13