Ja n 20 07 Heron ’ s Formula , Descartes Circles , and Pythagorean Triangles
نویسنده
چکیده
This article highlights interactions of diverse areas: the Heron formula for the area of a triangle, the Descartes circle equation, and right triangles with integer or rational sides. New and old results are synthesized. We first exploit elementary observations about circles to characterize an arbitrary triangle using three circles. A fourth circle brings a certain symmetry – the four radii are exactly the factors of the Heron area formula. The four equi-circles also play a role. When the triangle is a right triangle, the four circles combine into a single tangent cluster. Their centers form a rectangle, the system is reflection-congruent to its own dual set, and the dual circles correspond to the equi-circles (in-circle and ex-circles).
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