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Relating Trilinear and Tripolar Coordinates for a Triangle

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Given a triangle , the trilinear coordinates of a point are the signed distances to the extended sides. Denote the signed distances of to , and by , and , respectively. If and the incenter are in the same half-plane determined by a side, the signed distance to that side is positive; otherwise, it is negative.

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The tripolar coordinates of the point are its distances to the vertices of the triangle, given by , and .

The Conway triangle notation relates the sides to twice the area of the triangle, denoted by :

, , , .

These definitions imply the following formulas between the trilinear and tripolar coordinates:

, , .

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

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