Brocard inellipse
We want to know two things for each conic. Where does it fit into the grand scheme of triangle geometry, and how does it relate to other triangle structures?
This ellipse is part of a larger "Brocard picture, which is the inversive geometry of the triangle. This includes the Brocard circle, the Lemoine lattitude line (the inverse of the Brocard circle in the circumcircle) the Brocard points eO and wO (the foci of the ellipse), and e∞, w∞, the Beltrami points, their inversions to the circumcircle. and its . The six sides of these two triangles are tangent to the Brocard ellipse, all seen in the first picture below. This is discussed in the last chapter of Johnson.
Notation: a, b, c are reflections in an apolloninian circles. F is the focus of the Kiepert parabola, but has other significance as well. X is the crossing point, where the Brocard and Lemoine lines meet. It is the inverse of K in the circumcircle. e = ab and w = bc are rotations around the inversive plane, that produce both the Brocard points, when done to O, and the Beltrami points, when done to ∞. The inversive goemetry of the triangle, as expressed by Conway, is a very heady view indeed!
Figure: The Brocard geometry of the triangle ABC.
Points on the Brocard ellipse
Peter Moses finds these ETC points on the Brocard inellipse, center 39, perspector 6, (axis Brocard axis, foci eO and wO, the Brocard points)
{1015 a^2 (b-c)^2,
1017 {a^2 (-2 a+b+c)^2},
1977 a^4 (b-c)^2,
2028,2029 { -(a2 (a2b2+a2c2+b2c2) (a2b2-b4+a2c2-c4))±a2 (b2+c2) sqrt(a2 b2+a2 c2+b2 c2) sqrt((a4+b4+c4-b2c2-c2a2-a2b2) (b2c2+c2a2+a2b2))}}
Of the two points of intersection of the Brocard axis, OK with the Moses circle, X(2028) is the one nearer to X(3). Points X(2028) and X(2029) are the points of intersection of the Brocard axis and the asymptotes of the Kiepert hyperbola; see X(2039) and X(2040). For a description of the Moses circle and others, see the notes just above X(1662).
Inconics do not usually contain many points.
The Brocard ellipse is a strong object, so points on it occur in families, most are strong points and are solitary, There are not so many of these because inconics find it difficult to have many points on them. Quartile points come in 4's and are ofen the intersection of another inconic with the Brocard inconic. If a line intersects the elliplse, it produces a pair of point. On the Brocard ellipse is the interesting pair caused by the intersections of the Kiepert asymptotes with the Brocard meridian line at vertices of the Brocard ellipse. A similar phenomenon happens when the asymptotes of the Jerabek hyperbola go through the vertices of the MacBeath ellipse at its intersections with the Euler line.
| point | coordinates | mate | comments | |
| 1015 | a2(b-c)2 | 1500 a2(b+c)2 | 4 intersections with steiner inellipse | |
| 1017 | a2(b+c-2a)2 | |||
| 1977 | a4(b-c)2 | a4(b+c)2 | 4 intersections with pK inconic | |
| 2028,9 | Kiepert hyperbola asymptote intersections with Brocard inconic | |||
| pmS (not in ETC) | 1 of 4 intersections (the others are not centers) of the R inconic. | |||
Reation to other structures.
The Brocard inellipse is dual to the R ( = tK) circumellipse. This means that the dual of every point on Brocard is a line tangent to R-circumellipse.
Mates: Inconics meet 4 times, forming a quartile set of points. Of these, one is central and the other group of three, taken as a group, is also. The vertices of ABC joined to the corresponding intersection concur at the desmic mate of the central point. For example the desmic mate of 1015, formed as intersections of the Steiner inellipse with Brocard, is X1500, as shown in the above ilustration.
Mates: The weak points on a strong inconic are often the 4 intersections with another conic. For example the four X1015 points are the intersections of the Brocard and Steiner inellipses. Shown in this picture are the Brocard ellipse (red) and the R-inellipe, pK inellipse, and Steiner inellipses along with the quartile points formed by their intersections and the associated mated center.