Geometric Investigation 1584: Proving a Ratio with the Incircle, Tangency Points, and Angle Bisector

The Hidden Treasure of Triangles: Unveiling a Potentially Unexplored Geometric Relationship.

Test your problem-solving skills with this captivating geometry challenge featuring the incircle and angle bisector of triangle ABC. Designed to enhance critical thinking and logical reasoning, this problem is ideal for students, educators, and math enthusiasts alike.

Problem 1584 Statement

In triangle ABC, the incircle is tangent to sides AB, BC, and AC at points D, E, and F, respectively. Lines DE and BF intersect at point G, and BH is the internal angle bisector of angle B. Prove that the ratio of DG to GE is equal to the ratio of the product of segments AF⋅CH to AH⋅CF, i.e., \(\frac{DG}{GE}=\frac{AF\cdot CH}{AH\cdot CF}\).

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