Malfatti Circles - Radius Formula

Radius Formula

The radius of each of the three Malfatti circles may be determined as a formula involving the three side lengths a, b, and c of the triangle, the inradius r, the semiperimeter, and the three distances d, e, and f from the incenter of the triangle to the vertices opposite sides a, b, and c respectively. The formulae for the three radii are:


According to Stevanović (2003), these formulae were discovered by Malfatti and published posthumously by him in 1811.

Related formulae may be used to find examples of triangles whose side lengths, inradii, and Malfatti radii are all rational numbers or all integers. For instance, the triangle with side lengths 28392, 21000, and 25872 has inradius 6930 and Malfatti radii 3969, 4900, and 4356. As another example, the triangle with side lengths 152460, 165000, and 190740 has inradius 47520 and Malfatti radii 27225, 30976, and 32400.

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