Conics twinned with two special circles

Extract

Throughout the article we use areal (or barycentric) coordinates. The side lengths of BC, CA, AB are denoted as usual by a, b, c and for brevity and ease of reading we write a² = p, b² = q, c² = r. The symmedian point K then has coordinates (a², b², c² ) replaced by (p, q, r), it being the isogonal conjugate of the centroid, and the circumcentre O then has coordinates (a²(b² + c² − a²), b²(c² + a² − b²), c²(a² + b² − c²)) replaced by (p(q + r − p), q(r + p − q), r(p + q − r)). The construction of the triplicate ratio circle (also known as the Lemoine circle) and the Brocard circle is replicated but starting with the circumcentre O rather than the symmedian point K. It is found that the triplicate ratio circle is then replaced by a conic cutting internally each side of ABC twice and having centre X the midpoint of OK. The Brocard circle is replaced by a conic passing through O and K and also having centre X the midpoint of OK. These we define as the twin conics.