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In the diagram below, given a
semicircle of diameter AB, CD is a perpendicular to AB, circle
of diameter CD and semicircles of diameter AC and CB
respectively. If S1, S2, S3 , S4,
and S5 are the shaded areas, prove that: S1
+ S2 + S3 = S4 + S5.
.
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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. See: Archimedes' Book of
Lemmas
Proposition 4 Arbelos
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