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In the figure below, given a triangle
ABC, S1 is the area of contact triangle of the
incircle, Sa, Sb, and Sc are
the areas of the contact triangles of the excircles. Prove that:
 .
"A great discovery solves a great problem, but there is a grain
of discovery in the solution of any problem. Your problem may be
modest, but if it challenges your curiosity and brings into play
your inventive faculties, and if you solve it by your own means,
you may experience the tension and enjoy the triumph of
discovery. Such expert experiences at a susceptible age may
create a taste for mental work and leave their imprint on mind
and character for a lifetime." George Polya, 1944
HINTS:
1. See Problems
82,
83,
84.
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