Intersections at the Vertices of the Second Brocard Triangle Christopher Bradley

Article: CJB/2010/101 Intersections at the vertices of the Second Brocard Triangle Christopher Bradley

Abstract: If D, E, F are the ex-symmedian points it is proved that circles BCD, CAE, ABF intersect the Brocard circle at the circumcentre of ABC and at the vertices of the second Brocard triangle. A case of circular perspective is thereby determined.

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Fig. 1

Circles intersecting on the Brocard circle

1. Introduction

In Fig.1 O is the circumcentre of triangle ABC and K is the symmedian point. Points D, E, F are the ex-symmedian points (where the tangents at A, B, C to the circumcircle intersect). The circle on OK as diameter is the Brocard circle. In this short article we prove that circle BDC meets the Brocard circle at O and at U, the first vertex of the second Brocard tiangle, which also lies on line AKD. V and W, the second and third vertices of the second Brocard triangle are determined in similar fashion by circles CAE and ABF respectively. It is also shown that circles AVW, BWU, CUV pass through the same point P on the circumcircle.

2. The Brocard circle and the ex-symmedian points

The Brocard circle passes through O, K and the two Brocard points and has equation b2c2x2 + c2a2y2 + a2b2z2 – a4yz – b4zx – c4xy = 0. (2.1)

The three ex-symmedian points have co-ordinates D(– a2, b2, c2), E(a2, – b2, c2) and F(a2, b2, – c2).

3. Circles BCD, CAE, ABF

The equation of the circle BCD is b2c2x2 – a2(b2 + c2 – a2)yz + b2(a2 – b2) zx + c2(a2 – c2)xy = 0. (3.1)

The equations of circles CAE, ABF may now be written down from Equation (3.1) by cyclic change of x, y, z and a, b, c.

4. The Points of intersection

Circle BCD with equation (3.1) and the Brocard circle with equation (2.1) meet at O the circumcentre of triangle ABC and at the point U(b2 + c2 – a2, b2, c2). U is the first point of the second Brocard triangle (which is in perspective with the first Brocard triangle with perspector the centroid). The co-ordinates of the points V, W the intersections of circles CAE and ABF other than O with the Brocard circle may now be written down from those of U by cyclic change of x, y, z and a, b, c. They are, of course, the second and third vertices of the second Brocard triangle. From the co-ordinates of U, V, W, K, D, E, F it is also clear that AKUD, BKVE, CKWF are straight lines.

5. The intersection of circles AVW, BWU, CUV

The fact that circles BCU, CAV, ABW meet at the circumcentre O means that triangles ABC and UVW are in circular perspective (see [1], Article CJB/2010/19) which is a symmetric relationship means that circles AVW, BWU, CUV also have a common point P. The co- ordinates of P are (a2/(b2 – c2), b2/(c2 – a2), c2/(a2 – b2)). It may be checked that P lies on the circumcircle of triangle ABC.

Reference 1. C. J. Bradley, Article 19 of this series, Article: CJB/2010/19.

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