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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 circumcenters of
triangles ABC, DBE, FGE, and MGC
respectively, prove
that the quadrilateral OO1O2O3
is a parallelogram.
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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.
3. See Proposed Problem
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