## De Villiers points of a triangle

1) If I is the incentre of triangle ABC, and the incenters D, E and F are respectively constructed for AIB, BIC and CIA, then the lines AE, BF and CD are concurrent in the inner De Villiers point DEV1.
2) If K, L and M are the excentres of triangle ABC, and the incenters X, Y and Z are respectively constructed for AKB, BLC and CMA, then the lines AY, BZ and CX are concurrent in the outer De Villiers point DEV2.

#### De Villiers Points of a Triangle

To my great surprise, I accidentally in August 2008 found out that these two special points of a triangle have been named after me at WolframMathWorld as De Villiers Points in relation to a 1996 paper of mine in Mathematics & Informatics Quarterly. They are also referenced as Points 1127 and 1128 at the Encyclopedia of Triangle Centers But one is immediately humbled (and also slightly bemused) by the fact that today there are more than 3500 special points known in relation to the simple, elementary triangle, so mine are only two amongst thousands!

For a proof, read my 1996 article A Dual To Kosnita's Theorem which uses a generalization of the Fermat-Torricelli theorem.

My more recent AMESA July 2009 paper From the Fermat points to the De Villiers points of a triangle more fully describes the discovery and proofs involved, tracing the origin from the Fermat points of a triangle.

My meer onlangse Sept 2010 artikel Vanaf die Fermat punte na die De Villiers punte van 'n driehoek beskryf die ontdekking en betrokke bewyse meer volledig, asook die oorsprong en ontwikkeling vanaf die Fermat punte van 'n driehoek.

I was pleasantly amused to recently find in an online paper in Aug 2015 by two Bulgarian mathematicians that they called the triangles DEF and XYZ the 'De Villiers' triangles, and proposed several conjectures, discovered by computer, of the (inner) De Villiers triangle in relation to the Malfatti-Steiner point. Read the paper at Grozdev Dekov paper (2015).

## De Villiers Hyperbola

The three vertices of a triangle and the two De Villiers points lie on a hyperbola (drag any of the three vertices of the triangle to see dynamically).

Please enable Java for an interactive construction (with Cinderella).

Obviously the 5 points (A, B, C, D1 and D2) are sufficient to determine a conic, but to prove that it is always a hyperbola relies on an interesting result proven by John Silvester, King's College, London, that a conic through the 3 vertices of a triangle, and a point that is always interior, is a hyperbola. For an unedited outline of his proof, go to Hyperbola Proof. It could be interesting to explore if there are other special triangle points that lie on this hyperbola, and with over 3500 special points, there is some chance that some of them may lie on it.

## Collinearities

1) The first De Villiers point is collinear with points 174 and 481. See Collinearity 1.

2) The second De Villiers point is respectively collinear with a) points 164 and 174, b) 188 and 519 as well as c) 258 and 505. See X1128 and use this interactive C.a.R applet to construct these 6 related points and the respective 3 lines through them.

Go to the Encyclopedia of Triangle Centers for definitions of these related points.

## Further Investigations

1. What are the respective loci of the two De Villiers points if triangle ABC is inscribed in a given circle and BC is fixed so that only A moves along the circumference of the circle? Investigate using De Villiers Points Loci.

2. Does the 1st De Villiers point exist in elliptic (spherical) geometry? Investigate using De Villiers point 1 - elliptic.

3. Does the 1st De Villiers point exist in hyperbolic geometry? Investigate using De Villiers point 1 - hyperbolic.

4. Does the 2nd De Villiers point exist in elliptic (spherical) geometry? Investigate using De Villiers point 2 - elliptic.

5. Does the 2nd De Villiers point exist in hyperbolic geometry? Investigate using De Villiers point 2 - hyperbolic.

6. Now ask some questions yourself and explore further!

Created Michael de Villiers, 2009; modified 21 June 2016.