# Problem 96. Similar Triangles, Incenters, Parallelogram. High School, College

 In the figure below, given a triangle ABC, line DEF parallel to AC and line FGM parallel to AB. If O, O1, O2, and O3, are the incenters of triangles ABC, DBE, FGE, and MGC respectively, prove that the quadrilateral OO1O2O3 is a parallelogram. .

FACTS AND HINTS:

Geometry problem solving is one of the most challenging skills for students to learn. When a problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."

1. SIMILAR TRIANGLES:
Proposition:
Corresponding angles of similar triangles are congruent.

2. PROVING THAT LINES ARE PARALLEL:
Proposition:
Two lines are parallel if a pair of corresponding angles are congruent.
Proposition: Two lines are parallel if a pair of alternate interior angles are congruent.

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