HMMT 二月 2008 · TEAM2 赛 · 第 15 题

HMMT February 2008 — TEAM2 Round — Problem 15

[ 40 ] Let X be as in the previous problem. Let T be the point diametrically opposite to D on on the incircle of ABC . Show that 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 they pass 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 , tangent to all three sides. • The excircles of a triangle ABC are the three circles on the exterior the triangle but tangent to all three lines AB, BC, CA . 3 • 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 , respectively, the lines AX, BY, CZ are concurrent if and only if BX CY AZ · · = 1 . XB Y A ZB 4

解析

[ 40 ] Let X be as in the previous problem. Let T be the point diametrically opposite to D on on 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. This dilation must send the diameter DT to some the diameter of excircle that is perpendicular to 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 they pass 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 , tangent to all three sides. • The excircles of a triangle ABC are the three circles on the exterior the triangle but tangent to all three lines AB, BC, CA . • The orthocenter of a triangle is the point of concurrency of the three altitudes. 7 • Ceva’s theorem states that given ABC a triangle, and points X, Y, Z on sides BC, CA, AB , respectively, the lines AX, BY, CZ are concurrent if and only if BX CY AZ · · = 1 . XB Y A ZB 8