 # Distance between Gergonne Point and Fletcher Point

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

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

Define (the line is 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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