# Online Math: Geometry Problem 1217: Triangle, Circle, Excenter, Incenter, Angle Bisector, Cyclic Quadrilateral, Circumcircle, Tangent Line. Tiled background image: Intihuatana, Machu Picchu. Level: College, High School.

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 In a triangle ABC (see the figure), L and R are the excenters corresponding to BC and AC, respectively. BR cuts AL and AC at I and E, respectively. J is the incenter of triangle BEC. BJ cuts AL at P and EJ extended cuts BC, AL, and BL at Q, K, and M, respectively. Prove that (1) BICL, BIJK, BPQK, and ABLR are cyclic quadrilaterals; (2) AL is tangent to circumcircle of triangle BKM at T. This entry contributed by Sumith Peiris, Moratuwa, Sri Lanka.

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Last updated: May 21, 2016