Problem 345. Equal circles, Tangents, Concurrent lines, Hexagon,
Semiperimeter. Level: High School, College, SAT Prep.
The figure shows equal circles A, B,
and C. AA_{1}, AA_{2}, CC_{3}, and CC_{4}
are tangents to circle B, AA_{3}, AA_{4}, BB_{3},
and BB_{4} are tangents to circle C, and BB_{1},
BB_{2}, CC_{1}, and CC_{2} are tangents
to circle A. Points D, E, F, G, H, and M are the intersection
points of the tangents AA_{1} with BB_{1}, BB_{3}
with CC_{3}, AA_{4} with CC_{1}, AA_{2}
with BB_{2}, BB_{4} with CC_{4}, and AA_{3}
with CC_{2}, respectively. Prove that (1) The extension
of DG, EH, and FM are concurrent, (2) If s is the semiperimeter
of the hexagon ADBECF, then s = AD + BE + CF = BD + CE + AF.
Geometry problem solving
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 illstructured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."

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