Winter Camp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology yufei.zhao@gmail.com 1 Diameter of incircle A B C D X T Lemma 1. Let the incircle of triangle ABC touch side BC at D, and let DT be a diameter of the circle. If line AT meets BC at X , then BD = CX . A B C D E F X Y Z T Proof. Assume wlog that AB ≥ AC . Consider the dilation with center A that carries the incircle to an excircle. The line segment DT is the diameter of the incircle that is perpendicular to BC , and therefore its image under the dilation must be the diameter of the excircle that is perpendicular to BC . It follows that T must get mapped to the point of tangency between the excircle and BC . In addition, the image of T must lie on the line AT , and hence T gets mapped to X . Thus, the excircle is tangent to BC at X . It remains to prove that BD = CX . Let the incircle of ABC touch sides AB and AC at F and E, respectively. Let the excircle of ABC opposite to A touch rays AB and AC at Z and Y , respectively, 1