Geometry Problem 1616: Incenter and Segment Ratios in a Contact Triangle

Geometry Problem 1616: Triangle ABC with incircle, incenter I, and chord of tangency DE

Figure: Intersection of bisectors AI, CI, and radius extension FI with the chord of tangency DE.

Problem Statement:

In a triangle ABC, the incircle with center I is tangent to the sides AB, BC, and AC at the points D, E, and F, respectively.

The lines AI, FI, and CI intersect the chord of tangency DE at the points G, H, and M, respectively.

Given:
  • DM = 8
  • MH = 10
  • HG = 12

Find: The length of the segment GE.

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Key Properties for Solving:

  • • I is the incenter of ΔMFG.
  • • FI is the internal bisector of ∠MFG.
  • • Symmetry: GF = GD and MF = ME

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