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Problem 82. Area
of the Contact Triangle, Inradius, Circumradius.
Level: High School, SAT Prep, College |
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Given a triangle ABC of area S, the incircle of center I,
the inradius r,
and the circumradius R. If S1 is the area of the
contact triangle DEF, prove
that:
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"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:
See:
Proposed Problem 81.
CONTACT TRIANGLE:
The contact triangle of a triangle
ABC (figure above), also called the intouch triangle, is the
triangle DEF formed by the points of tangency of the incircle of
triangle ABC with triangle ABC.
AREA OF A TRIANGLE:
Semiperimeter and Inradius Formula
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