In the figure below, given a triangle
ABC, circumcircle C0, circumradius R, line DEF
parallel to AC and line FGM parallel to AB. C1, C2,
and C3, and R1, R2, and R3
are the circumcircles and circumradii of triangles DBE, FGE, and
MGC respectively, prove that: R // R1 // R2
// R3, and circles C0 and C1
are tangent at B, circles C1 and C2 are
tangent at E, circles C2 and C3 are
tangent at G, and circles C3 and C0 are
tangent at C.
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:
Corresponding angles of similar triangles are congruent.
2. PROVING THAT LINES ARE
Two lines are parallel if a pair of corresponding angles are
Proposition: Two lines are parallel if a pair of
alternate interior angles are congruent.
3. CIRCLES TANGENT:
In the figure above, circles C0 and C1 are
tangent internally if the line of center OO1 extended
passes through B.
Proposition: In the figure above, circles C1
and C2 are tangent externally if the line of center O1O2
passes through G.
All Problems |
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