HMMT 二月 2008 · TEAM2 赛 · 第 15 题

英文解析

15. [ 40 ] Let X be as in the previous problem. Let T be the point diametrically opposite to D onon the incircle of ABC . Show that A, T, X are collinear.
    Solution: Consider a dilation centered at A that carries the incircle to the excircle. Thisdilation must send the diameter DT to some the diameter of excircle that is perpendicularto BC . The only such diameter is the one goes through X . It follows that T gets carried to
    X . Therefore, A, T, X are collinear.
    Glossary and some possibly useful facts
    • A set of points is collinear if they lie on a common line. A set of lines is concurrent if theypass through a common point.
    • Given ABC a triangle, the three angle bisectors are concurrent at the incenter of the triangle.
    The incenter is the center of the incircle , which is the unique circle inscribed in ABC , tangentto all three sides.
    • The excircles of a triangle ABC are the three circles on the exterior the triangle but tangentto all three lines AB, BC, CA .
    • The orthocenter of a triangle is the point of concurrency of the three altitudes.
    • Ceva’s theorem states that given ABC a triangle, and points X, Y, Z on sides BC, CA, AB ,7
    respectively, the lines AX, BY, CZ are concurrent if and only if
    BX CY AZ
    · · = 1 .
    XB Y A ZB 8