Problem 346. Equal circles, Tangents, Hexagon, Semiperimeter.
Level: High School, College, SAT Prep.
The figure shows equal circles A, B,
and C. AA_{1} and CC_{1}
are tangents to circle B, AA_{2} and BB_{1} are
tangents to circle C, and BB_{2} 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_{2},
AA_{1}
with CC_{1}, BB_{1} with CC_{1}, BB_{1}
with AA_{2}, CC_{2} with AA_{2}, and CC_{2}
with BB_{2}, respectively. If s is the semiperimeter of
the hexagon DEFGHM, prove that s = DM+EF+GH = DE+FG+HM.
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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