Steiner's Construction
Although much of the early work on the Malfatti circles used analytic geometry, in 1826 Jakob Steiner provided the following simple synthetic construction.
A circle that is tangent to two sides of a triangle, as the Malfatti circles are, must be centered on one of the angle bisectors of the triangle (green in the figure). These bisectors partition the triangle into three smaller triangles, and Steiner's construction of the Malfatti circles begins by drawing a different triple of circles (shown dashed in the figure) inscribed within each of these three smaller triangles. Each pair of two of these three inscribed circles has two bitangents, lines that touch both of the dashed circles and pass between them: one bitangent is the angle bisector, and the second bitangent is shown as the red dashed line in the figure. Label the three sides of the given triangle as a, b, and c, and label the three bitangents that are not angle bisectors as x, y, and z, where x is the bitangent to the two circles that do not touch side a, y is the bitangent to the two circles that do not touch side b, and z is the bitangent to the two circles that do not touch side c. Then the three Malfatti circles are the inscribed circles to the three tangential quadrilaterals abyx, aczx, and bczy. The three bitangents x, y, and z cross the triangle sides at the point of tangency with the third inscribed circle, and may also be found as the reflections of the angle bisectors across the lines connecting pairs of centers of these incircles.
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