IncircleTangents2

Incircle tangents as trilinear polars (the equilateral case II)

[1] The tangents to the incircle of an equilateral triangle DEF are the trilinear polars (tripolars) with respect to DEF of the points at infinity.
[2] The tangents to the circumcircle of an equilateral triangle DEF are the trilinear polars of the points at infinity with respect to the anticomplementary triangle ABC of DEF.
[3] The two tangents L₁, L₂ defined in [1], [2] by the same point at infinity are parallel. The distance of these parallels is constant (independent of the direction of parallels defining the point at infinity) and equal to the altitude of DEF.

[1] and [2] are proved in IncircleTangents.html . The parallelity in [3] is a consequence of the fact that the two triangles DEF and ABC are anti-homothetic with respect to their common center and with homothety ratio 2. The constancy of the distance of these parallels is obvious from the fact that they are tangent to two concentric circles. From there also follows the claim on the value of this distance.

For a generalization of this property to arbitrary triangles see the discussion (especially [6]) in TriangleConics.html .

Incircle tangents as trilinear polars (the equilateral case II)

See Also