1. The Brocard circle and its isogonal conjugate locus

The equation of the Brocard circle is b2c2x2 + c2a2y2 + a2b2z2 – a4yz – b4zx – c4xy = 0 (1.1)

It is known, of course that this has OK as a diameter, where O is the circumcentre of triangle ABC and K is the symmedian point. Also, the two Brocard points W(1/b2, 1/c2, 1/a2) and W'(1/c2, 1/a2, 1/b2) lie on the circle, with the chord WW' perpendicular to OK.

The isogonal conjugate locus of the Brocard circle ha equation a2y2z2 + b2z2x2 + c2x2y2 – xyz(a2x + b2y + c2z). (1.2)

This contains the orthocentre H, the centroid G and again, the pair of Brocard points W and W'.

2. The six defining circles

The six defining circles of the Brocard points are: BCW: – b2x2 + (c2 – b2)xy + a2yz = 0, (2.1) CAW: b2zx – c2y2 + (a2 – c2)yz = 0, (2.2) ABW: – a2z2 + c2xy + (b2 – a2)zx = 0, (2.3) BCW': – c2x2 + (b2 – c2)zx + a2yz = 0, (2.4) CAW': b2zx – a2y2 + (c2 – a2)xy = 0, (2.5) ABW': – b2z2 + c2xy + (a2 – b2)yz = 0. (2.6)

3. The Second Brocard triangle PQR

The point P = CAW^ABW' has co-ordinates P(b2 + c2 – a2, b2, c2) and it is easily checked that P lies on the Brocard circle. The other two vertices of the Second Brocard triangle have co- ordinates Q(a2, c2 + a2 – b2, c2) and R(a2, b2, a2 + b2 – c2).

4. The Fourth Brocard triangle P'Q'R'

The point P' = ABW^CAW' has co-ordinates P'(a2, b2 + c2 – a2, b2 + c2 – a2) and it is easily checked that P' lies on the isogonal locus of the Brocard circle. In fact, P' is the isogonal conjugate of P. The other two vertices of the Fourth Brocard triangle have co-ordinates Q'(c2 + a2 – b2, b2, c2 + a2 – b2) and R'(a2 + b2 – c2, a2 + b2 – c2, c2) and these are the isogonal conjugates of Q and R respectively.

5. The First Brocard triangle LMN

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The equation of BW is x/a2 = z/b2, (5.1) and the equation of CW' is x/a2 = y/c2. (5.2)

The point L = BW^CW' has co-ordinates L(a2, c2, b2). The other two vertices M and N of the First Brocard triangle have co-ordinates M(c2, b2, a2) and N(b2, a2, c2).

6. The Third Brocard Triangle

The equation of CW is x/c2 = y/b2, (6.1) and the equation of BW' is x/b2 = z/c2. (6.2)

The point L' = CW^BW' has co-ordinates L'(b2c2, b4, c4). This point lies on the isogonal locus of the Brocard circle and indeed L' is the isogonal conjugate of L. The other two vertices M' and N' of the Third Brocard triangle have co-ordinates M'(a4, a2c2, c4) and N'(a4, b4, a2b2).

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