In the diagram below, the circles A and B intersect at C and
D. CE is tangent to circle A at C, CF is tangent to circle B at C,
circle G passes through the points C, E, F. If HM is tangent to circle G
at C, prove that C is the midpoint of HM.
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."