Three Tangents Theorem II: What is this about?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2018 Alexander Bogomolny

The notations used in the applet are reminiscent of those used in the See-Saw Lemma where the diagram is almost the same. There is just one difference: the point W present in the case at hand has been moved to infinity in the See-Saw theorem. The statement is actually simplified:

Let TA, TX, and TB be the tangents to a circle. Assume TA, TX meet in E, TB, TX meet in F and TA, TB meet in W. Then the three lines AF, BE and XW are concurrent.


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Note that if the three tangents are positioned so that the given circle is the incircle of ΔEFW, then the point of concurrency is none other than the Gergonne point of ΔEFW. The current statement adds an observation that, from similar considerations, concurrency is preserved if the incircle of a triangle is replaced by any of its excircles.

One observation both simplifies the proof and generalizes the statement. The Three Tangents Theorem is of projective character: it only talks of a second degree curve (circle), tangents to a curve and concurrency of three lines. All these properties are conserved under projective mappings. Therefore, for one, the theorem holds for all projective images of the circle, viz., all non-degenerate conics (ellipse, parabola and hyperbola.) Secondly, it is sufficient to prove the generalization for a circle, which has been done on another occasion for an excircle, and just above (with a reference to Gergonne's concurrency), for the incircle. Note also, that, "being inside a triangle" is not a projective property, there is no difference in projective geometry between in- and excircles. Thus establishing Gergonne's concurrency is sufficient to proving the theorem for a triangle formed by three tangents to a non-degenerate conic.

We have establsihed the Three Tangents Theorem:

Assume a non-degenerate conic is tangent to the sides QR, PR, and PQ of ΔPQR in points U, V, W, respectively. Then the lines PU, QV, and RW are concurrent.

This is a general form of Gergonne's theorem which, for ellipses, has a nice dynamic illustration.

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Copyright © 1996-2018 Alexander Bogomolny

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