Geometry Problem 1602: Find the Distance from the Semicircle’s Center to the Incenter

A classic puzzle with a twist: Can you uncover the hidden length using tangents and symmetry?

Problem Description

Let AB be the diameter of a semicircle with center O. A circle is inscribed, tangent to both the arc and the diameter, with center C. Tangents drawn to the circle centered at C, perpendicular to the diameter, intersect AB at points E and G. It is given that the distance from A to E is 3 units, and from B to G is 11 units. Determine the length of segment OC.

Diagram

Problem 1602: A circle is inscribed in a semicircle. Two perpendicular tangents intersect the diameter at points E and G. Given AE and BG, find the distance from the semicircle’s center O to the inscribed circle’s center C.

Hint & Strategy

  • Classical Theorem on Tangent Circles (Euclid’s Elements): If two circles are tangent to each other (either externally or internally), then their centers and the point of tangency lie on a straight line.
  • Euclid’s Theorem on Tangents from an External Point: "The two straight lines drawn from a point outside a circle to touch the circle are equal in length
  • The vertical tangents to the inscribed circle define a rectangle whose height is equal to the radius of the inscribed circle. Use this structure, along with the distances AE and BG, to find OC.

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