Explaining Some Collinearities Amongst Triangle Centres Christopher Bradley

Article: CJB/2011/138 Explaining some Collinearities amongst Triangle Centres Christopher Bradley

Abstract: Collinearities are preserved by an Affine Transformation, so that for every collinearity of triangle centres, there is another when an affine transformation has taken place and vice-versa. We illustrate this by considering what happens to three collinearities in the Euclidean plane when an affine transformation takes the symmedian point into the incentre.

9 pt circle

C'' A B'' Anticompleme ntary triangle F B' W' W E'  L M V O X Z E F' K 4 V' K'' Y G T H N'

D B U D'U' C

Triplicate ratio Circle

A''

A' Fig.1 Three collinearities amongst triangle centres in the Euclidean plane

1. When the circumconic, the nine-point conic, the triplicate ratio conic and the 7 pt. conic are all circles

The condition for this is that the circumconic is a circle and then the tangents at A, B, C form a triangle A'B'C' consisting of the ex-symmedian points A'(– a2, b2, c2) and similarly for B', C' by appropriate change of sign and when this happen AA', BB', CC' are concurrent at the symmedian point K(a2, b2, c2).

There are two very well-known collinearities and one less well-known, and we now describe them.

First there is the Euler line containing amongst others the circumcentre O with x-co-ordinate a2(b2 + c2 – a2), the centroid G(1, 1, 1), the orthocentre H with x-co-ordinate 1/(b2 + c2 – a2) and T, the nine-point centre which is the midpoint of OH. The equation of the Euler line is (b2 – c2)(b2 + c2 – a2)x + (c2 – a2)(c2 + a2 – b2)y + (a2 – b2)(a2 + b2 – c2)z = 0. (1.1)

Secondly there is the Brocard axis which serves as diameter for the 7 pt. circle, with O and K at opposite ends of the diameter. Other points on this axis are X, the centre of both the triplicate ratio circle and the 7 pt. circle, Z the midpoint of the segment joining the Brocard points Ω (1/b2, 1/c2, 1/a2) and Ω'(1/c2, 1/a2, 1/b2), as well as the fourth power point 4 with co-ordinates (a4, b4, c4). Z has co-ordinates (a2(b2 + c2), b2(c2 + a2), c2(a2 + b2)). The equation of the Brocard axis is b2c2(b2 – c2)x + c2a2(c2 – a2)y + a2b2(a2 – b2)z = 0. (1.2)

Thirdly there is the line joining K and G. This passes through K''(b2 + c2 – a2, c2 + a2 – b2, a2 + b2 – c2), which is the symmedian point of the anticomplementary triangle (which has ABC as its medial triangle) and the midpoint Y of KK'' (Y is X38 in the Encyclopaedia of Triangle Centres). Its co-ordinates are (b2 + c2, c2 + a2. a2 + b2). The equation of this axis is (b2 – c2)x + (c2 – a2)y + (a2 – b2)z = 0. (1.3)

2. What happens when K is replaced by the incentre I

This is brought about by choosing a circumconic (i.e. one that passes through A, B, C) such that the tangents at the vertices form a triangle with vertices at the excentres I1, I2, I3 and then the lines AI1, BI2, CI3 are concurrent at the incentre I. See Fig.2 below for the diagram.

From Article 137 what this amounts to is to replace a2, b2, c2 by a, b, c in areal co-ordinates of points and in equations of lines. We are therefore in a position to provide easily the comparison to the situations described in Section 1.

7 pt conic

F Feu W' W E' I2 Mi  V XZ F' K Na SpG I E 9 pt conic T V' ' Ge Inconic U D'U' C B D

Fig. 2 The image of the Euclidean plane when K is replaced by I

First consider the image of the Euler line. This contains the points Mi (the Mittenpunkt) with co- ordinates (a(b + c – a), b(c + a – b), c(a + b – c)), replacing O, the centroid G(1, 1, 1) and Ge (Gergonne’s point) with co-ordinates (1/(b + c – a), 1/(c + a – b), 1/(a + b – c)), replacing H and the midpoint of OH is now replaced by the midpoint of MiGe (still labelled T in Fig. 2). This, of course, is a known collinearity.

The equation of this axis is (b – c)(b + c – a)x + (c – a)(c + a – b)y + (a – b)(a + b – c)z = 0. (2.1)

Next consider the image of the Brocard axis. This has a diameter MiI and the 7 pt. conic has centre the image of X. The images of the Brocard points denoted by ω and ω' in Fig. 2 have co- ordinates (1/b, 1/c, 1/a) and (1/c, 1/a, 1/b) respectively. The midpoint of ωω' (still labelled Z) has co-ordinates (a(b + c), b(c + a), c(a + b)) and is X37 in the Encyclopaedia. It also contains K, the symmedian point, which is the image of 4, the fourth power point. The equation of this axis is bc(b – c)x + ca(c – a)y + ab(a – b)z = 0. (2.2)

Finally let us consider the image of the third line in Section 1. This is the line IG, which also contains Sp(the image of Y) with co-ordinates (b + c, c + a, a + b) and Na(the image of K'') with co-ordinates (b + c – a, c + a – b, a + b – c). Note that Sp is the midpoint of INa. The equation of this axis is (b – c)x + (c – a)y + (a – b)z = 0. (2.3)

Admittedly there is nothing new in this short article, but it serves to illustrate that there is a system and reason for many of the collinearities appearing in the Encyclopaedia of Triangle Centres.

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