 # Distance between Incenter and Fletcher Point

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Given a triangle , draw the incircle. The tangent points , , define the contact triangle. Let lines and intersect at , and at , and at . The points , , are collinear and form the perspectrix of the contact triangle, also called the Gergonne line, shown in orange.

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In the Soddy construction, the triangle vertices form the centers of three mutually tangent circles. There are two circles tangent to those three original circles, the inner and outer Soddy circles, with centers called the Soddy points (not labeled). The two Soddy points define the Soddy line (red), which includes the incenter of .

The Gergonne line and the Soddy line are perpendicular and intersect at the Fletcher point .

Let , , be the circumradius, inradius and semiperimeter of , respectively.

Define (shown in red).

Then .

You can drag the points , and .

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Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA

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Minh Trinh Xuan

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