Proposition

The figure below shows a right triangle ABC (angle B = 90 degree) with the altitude BH. The incircle I is tangent to AB, BC, and AC at D, F, and H, respectively. The excircle E corresponding to BC is tangent to BC at G. DG and BH meet at J. Prove that the triangle JBF is an isosceles triangle (BJ = BF).
 

 

See solution below

 

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Geometric Art: Isolines of Problem 1414

 

See also: Typography and poster of problem 1408.