HMMT 二月 2015 · 几何 · 第 3 题

HMMT February 2015 — Geometry — Problem 3

Let ABCD be a quadrilateral with ∠ BAD = ∠ ABC = 90 , and suppose AB = BC = 1, AD = 2. The circumcircle of ABC meets AD and BD at points E and F , respectively. If lines AF and CD meet at K , compute EK .

解析

Let ABCD be a quadrilateral with ∠ BAD = ∠ ABC = 90 , and suppose AB = BC = 1, AD = 2. The circumcircle of ABC meets AD and BD at points E and F , respectively. If lines AF and CD meet at K , compute EK . √ 2 Answer: Assign coordinates such that B is the origin, A is (0 , 1), and C is (1 , 0). Clearly, E 2 1 1 is the point (1 , 1). Since the circumcenter of ABC is ( , ), the equation of the circumcircle of ABC is 2 2 1 1 1 6 3 2 2 ( x − ) + ( y − ) = . Since line BD is given by x = 2 y , we find that F is at ( , ). The intersection 2 2 2 5 5 √ 3 1 2 of AF with CD is therefore at ( , ), so K is the midpoint of CD . As a result, EK = . 2 2 2 This is in fact a special case of APMO 2013, Problem 5, when the quadrilateral is a square.