Geometry Problem 1603: Can You Find the Hidden Distance AG?

One square. One circle. One secant from point C. A square inside a circle reveals a geometric mystery.

Problem Description

In a square with side length one, a line is drawn from vertex C. It intersects the incircle at points E and F, and side AD at point G. Segment EF is a side of a square inscribed in the circle. Find the distance from point A to point G.

Diagram

Problem 1603: Square ABCD with incircle and a secant from C intersecting the circle and side AD, forming points E, F, and G.

Key Concepts to Use

  • Properties of squares and incircles.
  • The side of a square inscribed in a circle.
  • Similar triangles or Euclidean ratios.
  • Visual reasoning and segment relations in the square.

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