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### **Problem 5**:
1. **Given**: A circle with center \( O \) and radius \( R \), and a point \( M \) outside the circle such that \( OM = 2R \). From point \( M \), the tangents \( MC \) and \( MD \) touch the circle at points \( C \) and \( D \), respectively, and \( AB \) is the common chord.
2. **Task a**: Prove that the triangle \( MCKO \) is cyclic and determine the center of the external circle.
3. **Task b**: Show that \( MC^2 = MA \cdot MB \).
4. **Task c**: Given \( AB = R\sqrt{3} \), find \( MA \) based on \( R \).

### **Problem 6**:
1. **Given**: Point \( M \) lies outside a circle with center \( O \) and radius \( R \), and point \( M \) lies on two tangents to the circle, \( MB \) and \( MC \), where \( B \) and \( C \) are the points of tangency.
2. **Task a**: Prove that triangle \( MBOC \) is cyclic.
3. **Task b**: Show that the line from \( M \) to \( D \) does not pass through the center \( O \) (where \( D \) is on the line segment between \( A \) and \( M \)), and that \( MB^2 = MA \cdot MD \).
4. **Task c**: Let \( K \) be the midpoint of segment \( AB \). Prove that the angle \( \angle BEC \) is equal to \( \angle BOM \).

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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.

Explore how to prove that a given quadrilateral is cyclic, apply the Power of a Point theorem, and solve for a segment length in a geometric problem involving a circle, tangents, and secants. This problem is ideal for high school students studying advanced circle geometry.

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