Power of a Point, Cyclic Quadrilaterals, and Some Useful Geometry Lemmas George Arzeno June 6, 2011
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Some Tips for Solving Geometry Problems • You will probably mark a lot of angles and/or lengths in your diagram, so a small diagram will get really messy. Draw a large one! • Draw your diagram precisely using a ruler and compass. This will help you conjecture about relationships between the lines, segments, and angles within it. • When you see a problem for the first time, play around with it for a while. Mark angles and look for relationships such as similar triangles and cyclic quadrilaterals. • If you are stuck, write a list of the things you know and another list of the things you want. Keep them separate to avoid circular arguments. In the list of things you know, look for what you have not used yet and see how it relates to the problem. • If you are still stuck, take a step back and look at the big picture. Think of which strategies you might use to solve the problem. For example, if you see a lot of lengths and circles, that means power of a point might be used at some point. • If you are in an olympiad and you have been stuck on a problem for a long time, work on another problem for a while. If you switch, your subconscious keeps thinking about the problem you cannot solve. When you go back to the problem you were stuck on, you might have an idea about what to do. • In many problems you need to construct some points or lines other than those given in the problem. Don’t just draw them at random. Try to think of a reason why defining a certain new point will be helpful. Use your intuition!
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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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Olympiad Level Problems
(IMO 1995 P1) Let A, B, C, D be four distinct points in a line, in that order. The circles with diameters AC and BD intersect at X and Y . The line XY meets BC at Z. Let P be a point on the line XY other than Z. The line CP intersects the circle with diameter AC at C and M , and the line BP intersects the circle with diameter BD at B and N . Prove that the lines AM , DN , XY are concurrent. (IMO 2009 P2) Let ABC be a triangle with circumcenter O. Let P and Q be interior points of the sides CA and AB, respectively. Let K, L, and M be the midpoints of the segments BP , CQ, and P Q, respectively, and let Γ be the circle passing through K, L, and M . Suppose that the line P Q is tangent to the circle Γ. Prove that OP = OQ. (IMO 2008 P1) Let H be the orthocenter of an acute triangle ABC. The circle ΓA is centered at the midpoint of BC and passes through H. Suppose ΓA intersects the sideline BC at points A1 and A2 . Define B1 , B2 , C1 , C2 similarly. Prove that the six points A1 , A2 , B1 , B2 , C1 , C2 are concyclic. (IMO 2006 P1) Let ABC be a triangle with incenter I. A point P in the interior of the triangle satisfies 6 P BA + 6 P CA = 6 P BC + 6 P CB. Show that AP ≥ AI, with equality holding if and only if P = I. (IMO 2004 P1) Let ABC be an acute-angled triangle with AB 6= AC. The circle with diameter BC intersects the sides AB and AC at M and N , respectively. Denote by O the midpoint of the side BC. The bisectors of the angles 6 BAC and 6 M ON intersect at R. Prove that the circumcircles of the triangles BM R and CN R have a common point lying on the side BC. (USAMO 2009 P1) Given circles ω1 and ω2 intersecting at points X and Y , let l1 be a line through the center of ω1 intersecting ω2 at points P and Q, and let l2 be a line through the center of ω2 intersecting ω1 at points R and S. Prove that if P , Q, R, and S lie in a circle, then the center of this circle lies on the line XY . (USAMO 2010 P1) Let AXY ZB be a convex pentagon inscribed in a semicircle of diameter AB. Denote by P Q, R, and S the feet of the perpendiculars from Y onto the lines AX, BX, AZ, BZ, respectively. Prove that the acute angle formed by lines P Q and RS is half the size of 6 XOZ, where O is the midpoint of segment AB. (IBERO 2009 P4) Let ABC be a triangle with incenter I, let P be the intersection of the external bisector of angle A and the circumcircle of ABC, and let J be the second intersection of P I with the circumcircle of ABC. Show that the circumcircles of triangles JIB and JIC are tangent to lines IC and IB, respectively. (ISL 2007 G2) Denote by M the midpoint of side BC in an isosceles triangle ∆ABC with AC = AB. Take a point X on the smaller arc M A of the circumcircle of triangle ∆ABM . Denote by T the point inside of angle BM A such that 6 T M X = 90 and T X = BX. Prove that 6 M T B − 6 CT M does not depend on the choice of X.
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