Power of a Point, Cyclic Quadrilaterals, and Some Useful Geometry Lemmas George Arzeno June 6, 2011

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Warm-Up Problems

Altitudes AD, BE, and CF of acute triangle ∆ABC intersect at H. Show that 6 BF D = 6 ACB. Let ω1 and ω2 be two circles that intersect in A and B. Suppose XY is a common tangent between the two circles, with X in ω1 and Y in ω2 . Let D be the intersection of XY with AB. Prove DX = DY .

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AIME Level Problems

In ∆ABC, 6 ABC = 50 and 6 ACB = 70. Let D be the midpoint of side BC. A circle is tangent to BC at B and is also tangent to segment AD. This circle intersects AB again at P . Another circle is tangent to BC at C and is also tangent to segment AD. This circle intersects AC again at Q. Find 6 AP Q. (AIME 2011 P4) In triangle ABC, AB = 125, AC = 117, and BC = 120. The angle bisector of angle A intersects BC at point L, and the angle bisector of angle B intersects AC at point K. Let M and N be 1

the feet of the perpendiculars from C to BK and AL, respectively. Find M N . (AIME 2009 P5) Triangle ABC has AC = 450 and BC = 300. Points K and L are located on AC and AB respectively so that AK = CK, and CL is the angle bisector of angle C. Let P be the point of intersection of BK and CL, and let M be the point on line BK for which K is the midpoint of P M . If AM = 180, find LP .

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