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CIRCLES ALL THE TIME
MBHS Math Team
December 22, 2010
Power of a Point
• Product of whole and exterior is constant if point is outside circle.
• Product of two parts of the chords is constant.
Point A is external to circle ω. P is on ω such that AP is tangent to ω. AO is drawn and
intersects ω at B and C with AB < AC. If AP = 2 · AB, compute AO in terms of AB.
Point B is on circle O and point P is external to O such that OP is perpendicular to
PA
.
OB and OP = 2 · OP . BP intersects circle O at A. Compute AB
Let A, B, C, and D be points on a circle such that AB = 11 and CD = 19. Point P
is on segment AB with AP = 6, and Q is on segment CD with CQ = 7. The line through
P and Q intersects the circle at X and Y . If P Q = 27, find XY .
Angles
Important theorem: inscribed angle is 21 the intercepted arc.
Resulting theorems: angles that intercept the same (or equal) arcs are congruent; the
hypotenuse of a right triangle is the diameter of the circumscribed circle,
Let ABC be a triangle with angle A = 45 degrees. Let P be a point on side BC with
P B = 3 and P C = 5. Let O be the circumcenter of ABC. Determine the length OP .
A regular polygon with 2n sides is inscribed in a circle, and a point on the circle is chosen. The sum of the squares of the distances from this point to the vertices of the polygon
is S. Compute S in terms of n.
In a triangle ABC, take point D on BC such that DB = 14, DA = 13, DC = 4, and
the circumcircle of ADB is congruent to the circumcircle of ADC. What is the area of
triangle ABC?
Tangents
Important facts: equal tangents; tangents are perpendicular to the radius.
Circle ω has radius 5 and is centered at O. Point A lies outside ω such that OA = 13. The
two tangents to ω passing through A are drawn, and points B and C are chosen on them
(one on each tangent) such that line BC is tangent to ω and ω lies outside triangle ABC.
Compute AB + AC given that BC = 7.
Quadrilateral ABCD has a circumscribed circle. Derive an equation relating side lengths
AB, BC, CD, and AD.
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CIRCLES ALL THE TIME
MBHS Math Team
December 22, 2010
Suppose ABCD is an isosceles trapezoid in which AB k BC. Two mutually externally
tangent circles ω1 and ω2 are inscribed in ABCD such that ω1 is tangent to AB, BC, and
CD while ω2 is tangent to AB, AD, and CD. Given that AB = 1 and CD = 6, compute
the radius of either circle.
The internal tangent of two circles is 7 and the external tangent is 11. Compute the product
of the radii.
Cyclic Quadrilaterals
Ways to show: opposite angles are supplementary; converse of the power of the point
formula.
Ptolemy’s theorem: AC · BD = AD · CD + AD · BC
Quadrilateral ABCD has diagonals intersecting in E such that AE = 2, BE = 5, CE = 10,
. Show that ABCD is cyclic. Then find AB.
DE = 4, and BC = 15
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Equilateral triangle ABC is inscribed in a circle and a point P is on arc BC. Prove that
P A = P B + P C.
Regular pentagon ABCDE is inscribed in a circle and point P is on arc BC. Prove that
P A + P D = P B + P C + P E.
Challenges
Points A, B, C, and D lie on circle ω. BA is extended through A and DC is extended
through C, and the two segments meet at P . AD intersects BC at E. If P A = 2, P B = 8,
and P C = 4, compute the ratio of the area of quadrilateral P AEC to the area of triangle
BAE.
Rectangle ABCD is inscribed in circle O with AB = 6 and AD = 8. Point E is on
AD such that AE = 6. BE is extended through E to intersect O at F . Compute AF .
Prove Ptolemy’s theorem.
Problem credits: HMMT, ARML, Challenging Problems in Geometry by Posamentier and
Salkind, Walsteins Advanced Geometry homework
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