HMMT 二月 2015 · 几何 · 第 3 题

英文解析

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 CDmeet at K , compute EK .
   √
   Answer: Assign coordinates such that B is the origin, A is (0 , 1), and C is (1 , 0). Clearly, E2
   1 12
   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.