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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
92
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