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Warm-upTangent circlesAngles inside circlesPower of a pointGeometryCirclesMisha LavrovARML Practice 12/08/2013Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointProblemsSolutionsWarm-up problems1A circular arc with radius 1 inch is rocking back and forth on aflat table. Describe the path traced out by the tip.2A circle of radius 4 is externally tangent to a circle of radius 9.A line is tangent to both circles and touches them at points Aand B. What is the length of AB?Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointProblemsSolutionsWarm-up solutions1A horizontal line 1 inch above the table.2Let X and Y be the centers of the circle. Lift AB to XZ as inthe diagram below.YXZABThen XY 4 9 13, YZ 9 4 5, and XZ AB. BecauseXZ 2 YZ 2 XY 2 , we get XZ AB 12.Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointProblemSolutionChallengeMany tangent circlesShown below is the densest possible packing of 13 circles into asquare. If the radius of a circle is 1, find the side length of thesquare.Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointProblemSolutionChallengeMany tangent circles: SolutionpABC is equilateral with sidep length 2, so C is 3 units above A.ACD is isosceles, so D isp 3 units above C , and finally Epis 2 unitsabove D. So AE 2 2 3, and the square has side 4 2 3.EDCABMisha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointProblemSolutionChallengeMore tangent circlesFor a real challenge, try eleven circles. Yes, two of those are loose.The solution is approximately but not quite 7; you can use this tocheck your answer.Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointFactsProblemsAngles inside circles: solutionsFacts about anglesDefinition. We say that the measure of an arc of the circle is themeasure of the angle formed by the radii to its endpoints.AA2A1CBBC2C1Ù.Theorem 1. If A, B, and C are points on a circle, ABC 12 ACTheorem 2. If lines from point ³B intersect a circle at A1 , A2 andÚC1 , C2 , A1 BC1 A2 BC2 12 AÚ2 C2 A1 C1 .Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointFactsProblemsAngles inside circles: solutionsAngles inside circles: problems1A hexagon ABCDEF (not necessarily regular) is inscribed ina circle. Prove that A C E B D F .2In the diagram below, PA and PB are tangent to the circlewith center O. A third tangent line is then drawn,interesecting PA and PB at X and Y . Prove that the measureof XOY does not change if this tangent line is moved.BOYPXMisha LavrovAGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointFactsProblemsAngles inside circles: solutionsSolutions12³ Ù CDÙ DEÙ EFØ and similarly forBy Theorem 1, A 12 BCother angles. If we add up such equations for A C E ,Ù BCÙ CDÙ DEÙ EFØ AFÙ 360 , and the samewe get ABfor B D F .Let Z be the point at which XY is tangent to the circle. Then4XAO and 4XZO are congruent, because AO ZO,XO XO, and XAO XZO 90 . ZOX XOA, andboth are equal to 12 ZOA. Similarly, ZOY YOB, andboth are equal to 21 ZOB. Adding these together, we get XOY 12 AOB, which does not depend on the position ofthe tangent line.Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointFactsProblemsSolutionsPower of a pointTheorem 3. Two lines through a point P intersect a circle at pointsX1 , Y1 and X2 , Y2 respectively. Then PX1 · PY1 PX2 · PY2 .X2X1Y1PY1Y2X1PX2Y2One way to think about this is that the value PX · PY you get bychoosing a line through P does not depend on the choice of line,only on P itself. This value is called the “power of P”.Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointFactsProblemsSolutionsPower of a point: problems1Assume P is inside the circle for concreteness. Prove that4PX1 X2 and 4PY2 Y1 are similar. Deduce Theorem 3.2Two circles (not necessarily of the same radius) intersect atpoints A and B. Prove that P is a point on AB if and only ifthe power of P is the same with respect to both circles.3Three circles (not necessarily of the same radius) intersect at atotal of six points. For each pair of circles, a line is drawnthrough the two points where they intersect. Prove that thethree lines drawn meet at a common point, or are parallel.Misha LavrovGeometry
Warm-upTangent circlesAngles inside circlesPower of a pointFactsProblemsSolutionsPower of a point: solutions1 X1 PX2 and Y1 PY2 are vertical, and therefore equal.Û PX1 X 2 and PY2 Y1 both intercept arc X2 Y1 , so they areboth equal by Theorem 1. So the triangles are similar.PY21Therefore PXPX2 PY1 , and we get Theorem 3 bycross-multiplying.2If P is on AB, the line PA intersects either circle at A and B,so the power of P is PA · PB.But if P is not on AB, the line PA does not pass through B: itintersects one circle at X and another at Y , so the power of Pis PA · PX for one circle and PA · PY for the other.3If any two lines intersect, the point of intersection will havethe same power for all three circles, so it lies on all three linesby the previous problem.Misha LavrovGeometry
Misha Lavrov Geometry. Warm-up Tangent circles Angles inside circles Power of a point Facts Problems Solutions Power of a point: solutions 1 \X 1PX 2 and\Y 1PY 2 arevertical,andthereforeequal. \PX 1X2and\PY 2Y 1 bothinterceptarcÛX 2Y 1,sotheyare bothequalbyTheorem1. Sothetrianglesaresimilar.
