Math Problem Statement

Given a circle with center O and radius R, and a point M outside the circle such that OM = 2R. Tangents MC and MD are drawn from M to the circle, touching it at C and D respectively. A secant MAB is also drawn, intersecting the circle at A and B. Prove that quadrilateral MCKO is cyclic, establish the equation MC^2 = MA * MB, and determine MA if AB = R√3.

Solution

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Problem 5:

  1. Given: A circle with center OO and radius RR, and a point MM outside the circle such that OM=2ROM = 2R. From point MM, the tangents MCMC and MDMD touch the circle at points CC and DD, respectively, and ABAB is the common chord.
  2. Task a: Prove that the triangle MCKOMCKO is cyclic and determine the center of the external circle.
  3. Task b: Show that MC2=MAMBMC^2 = MA \cdot MB.
  4. Task c: Given AB=R3AB = R\sqrt{3}, find MAMA based on RR.

Problem 6:

  1. Given: Point MM lies outside a circle with center OO and radius RR, and point MM lies on two tangents to the circle, MBMB and MCMC, where BB and CC are the points of tangency.
  2. Task a: Prove that triangle MBOCMBOC is cyclic.
  3. Task b: Show that the line from MM to DD does not pass through the center OO (where DD is on the line segment between AA and MM), and that MB2=MAMDMB^2 = MA \cdot MD.
  4. Task c: Let KK be the midpoint of segment ABAB. Prove that the angle BEC\angle BEC is equal to BOM\angle BOM.

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Math Problem Analysis

Mathematical Concepts

Circle Geometry
Cyclic Quadrilaterals
Power of a Point
Tangents and Secants

Formulas

Power of a Point: MC^2 = MA * MB
Chord Length Formula

Theorems

Cyclic Quadrilateral Theorem
Power of a Point Theorem
Intersecting Secant-Tangent Theorem

Suitable Grade Level

Grades 10-12