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Given a triangle ABC, the altitudes
AD, BE, and CF, the orthocenter H. If a,b,c,d,e, and f are the
inradii of triangles AFH, BDH, CEH, CDH, AEH, and BFH
respectively, prove that a.b.c = d.e.f
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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:
Triangle
Proposition: The sum of
the measures of the three angles of a triangle is 180.
Altitude is the perpendicular line segment from
one vertex to the line that contains the opposite side.
Similar Triangles
are triangles whose corresponding angles are congruent and whose
corresponding sides are in proportion.
Proposition. Triangle Similarity AA. If two
angles of one triangle are congruent to two angles of another
triangle, the two triangles are similar.
Proposition: If two triangles
are similar, then the ratio of any two corresponding segments
(such as altitudes, medians, inradii, or angle bisectors) equals
the ratio of any two corresponding sides.
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