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Given a triangle ABC of area S, the
incircle of center I and inradius r. If AC = b and AB = c, prove
that area of triangle DIE
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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:
CIRCLE is the set of all points in a
plane that are at the same distance from a fixed point
called the center.
Tangent of a circle is a
line that touches the circle at one and only one point no
matter how far produced.
Proposition. If a line is
tangent to a circle, it is perpendicular to a radius at the
point of tangency.
INCENTER of a triangle
Proposition: The bisectors
AD, BF and CE of the angles of a triangle ABC meet in a
point I, which is equidistant from the sides of the
triangle.
The incircle is the inscribed
circle of a triangle. The center of the incircle is called
the incenter, and the radius of the circle is called the
inradius.
PERPENDICULAR LINES AND ANGLES
Proposition: Two angles
are congruent or supplementary if their sides are
respectively perpendicular to each other.
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).
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