Geometry Problem 1604: Can You Find FG?

Explore this elegant problem involving two circles, a midpoint, a tangent, and an unknown segment.

Problem Description

Two circles intersect at A and B. A line through A intersects the first circle at C and the second at D, with A as the midpoint of CD. A tangent from C to the first circle intersects the second circle at E and F (F farther from C). Line FB extends to intersect the first circle at G. If CF = 6 and DG = 7, find FG.

Diagram

Problem 1604: Diagram showing two intersecting circles, midpoint A on segment CD, tangent from C, points E and F on second circle, and line FB meeting first circle at G.

šŸ§© Suggested Strategies and Key Ideas

  • Intersecting Circles: Two circles intersect at points A and B; AB is a common chord.
  • Midpoint on Transversal: A is the midpoint of CD, where C and D lie on different circles.
  • Tangent Line: The line from C is tangent to the first circle.
  • Cyclic Quadrilateral: Points like C, A, B, G may form a cyclic figure.
  • Power of a Point: Apply at C or F using tangentā€“secant relationships.
  • Similarity of Triangles: Triangle pairs may reveal proportional segments.
  • Angle Chasing: Use circle and tangent angles to establish equal angles.
  • Similarity of Triangles: Identify similar triangles to relate sides and angles, especially involving CF and DG.
  • Extension and Intersection: Line FB extended intersects the first circle at G.
  • Segment Lengths: CF = 6, DG = 7; use to find FG.

šŸ§­ Read more, explore further

Geometry Problems Open Problems Visual Index All Problems Circle Intersecting Circles Midpoint Tangent Line Secant line Similarity Triangle Metric Relation View or Post a solution