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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
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. 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 = ˝ (side1 × side2) × sine(included angle).
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