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

Explanation

The notations used in the applet are reminiscent of a different result where the diagram is almost the same. There is just one difference: the point W there has been moved to infinity. The statement is actually simplified:

(1)

Let TA, TX, and TB be the tangents to the circumcircle of AXB at vertices A, X, and B respectively. Assume that BX and AX produced beyond X intersect TA and TB in Y and Z, respectively. Assume also that TX meets TA in U and TB in V. Then the three lines TX, UV, and AB are either parallel or concurrent.

The three lines could be parallel iff X is the midpoint of the arc AXB. In this case, the lines are parallel because of the symmetry of the configuration. However, the above statement is conspicuously asymmetric with regard to the vertices of the triangle. The vertex X has been singled out in that the corresponding tangent T_X plays a role different from that of the other two, T_A and T_B, which prompts a second look at the diagram.

In fact, the vertices of AXB, besides P, there are other two points through which pass three lines: Y and Z. It's not hard to see that those points play the role of P under permutation of the vertices. The three points P, Y, Z have a few things in common:

1. P, Y, Z are collinear.

2. Each is found at the intersection of one of the tangents with the opposite side line of AXB.

Thus we come up with a reformulation in which the vertices of the triangle play similar roles:

(2)	Let TA, TX, and TB be the tangents to the circumcircle of AXB at vertices A, X, and B respectively. Let points Y, P, and Z denote the points of intersection of the tangents with the opposite side lines of AXB. Then the three points P, Y, Z are collinear.

This is the statement I am going to prove.

Proof

For the proof note that regardless of how the points P, Y, Z have been obtained, the lie one on each side line of AXB. We are to prove that the points are collinear. This calls for the converse of Menelaus' theorem. According to the latter, (2) will follow from

where the segments involved are thought to be signed.

To deal with the first ratio, AP/BP, we note a pair of similar triangles. In AXP and XBP that share angle at P, we also have

(4)	XAP = BXP.

For, XAP (= XAB) subtends the same arc as is enclosed into by BXP formed by chord AB and the tangent T_B at B. From the similarity of the triangles we derive:

from which it follows that

Similarly we obtain

so that (3) follows from (5_P), (5_Y), (5_Z).

The above configuration of a triangle and the tangents to its circumcircle at the vertices is quite remarkable in many other respects. The triangle formed by the lines T_A, T_X and T_B is known as the tangential (or contact) triangle of the reference AXB. By definition, the circumcircle of AXB is simultaneously the incircle of its tangential triangle. Since, a symmedian through a vertex passes through the point of intersection of the tangents to the circumcircle at the other two vertices, its Lemoine point also serves as the Gergonne point of the tangential triangle.

The line YZP is known as the Lemoine line of AXB. It's the polar of the Lemoine point with respect to the circumcircle. This is the Gergonne line of the tangential triangle.

Symmedians through vertices of AXB divide the opposite sides in exactly the proportions (5_P), (5_P), (5_Z). For this reason, the tangents T_A, T_X and T_B are often called external symmedians of AXB. The points of intersection of the internal and external symmedians through the same vertex with the opposite side are harmonic conjugates of each other with respect to the other two vertices.

The points Y, Z, and P are known as Nobbs' points of AXB.

Finally, the Three Tangents Theorem has a more general formulation.

References

1. F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991, pp. 1175-1177
2. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 151-153
3. R. Honsberger, In Pólya's Footsteps, MAA, 1997, pp. 175-176

The Symmedians

1. Symmedian and 2 Antiparallels
2. Symmedian and Antiparallel
3. Symmedian in a Right Triangle
4. Nobbs' Points and Gergonne Line
5. Three Tangents Theorem
6. A Tangent in Concurrency
7. Symmedian and the Tangents
8. Bride's Chair
9. Star of David
10. Ceva's Theorem
11. The Many Ways To Construct a Triangle
12. Concyclic Circumcenters: A Dynamic View
13. Concyclic Circumcenters: A Sequel