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In the figure below, given a triangle
ABC, Si is the area of the contact or intouch
triangle DEF, Se is the area of the extouch triangle
GHM. 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. The contact triangle of a triangle ABC, also
called the intouch triangle or Gergonne triangle,
is the triangle DEF formed by the points of tangency of the
incircle of triangle ABC with triangle ABC.
2.
The extouch triangle of a triangle ABC is the
triangle GHM formed by the points of tangency of the
triangle ABC with its excircles.
3. TANGENT TO A CIRCLE
Proposition.
Two tangent segments to a circle from an external point are
congruent.
4. Semiperimeter s, Side and
Incircle Formula
5. Semiperimeter s, Side and
Excircle Formula
6. AREA OF A TRIANGLE:
Proposition:
The area of a triangle equals
one-half the product of the length of a side and the length of
the altitude to that side.
Side Angle Side Formula: The
SAS formula = ½ (side1 × side2) × sine(included angle).
6. See Problems
82,
85.
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