Geometry Problem 1569: Proving the Length of EG Equals the Diameter of the Incircle of Triangle BDE in Isosceles Triangle ABC with Inscribed Square CDEF. A High School and College Geometry Challenge

In an isosceles triangle ABC with AB=BC, a square CDEF is constructed with D on BC and E on AB. The side EF intersects AC at G. Prove that the length of EG equals the diameter of the incircle of triangle BDE.

Academic Levels: Suitable for High School and College Mathematics Education

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Key Definitions and Descriptions

Vocabulary	Description
Isosceles Triangle	A triangle with two sides of equal length.
Isosceles Triangle ABC	An isosceles triangle where AB = BC.
Square CDEF	A square constructed such that vertex D lies on BC and vertex E lies on AB.
Point G	The intersection point of side EF of the square with AC of the triangle.
EG	The segment from E to G, which we need to prove is equal to the diameter of the incircle of triangle BDE.
Incircle	The largest circle that fits inside a triangle, touching all three sides.
Diameter of the Incircle	Twice the radius of the incircle, which is the circle inscribed within triangle BDE.

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