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X(99), X(511)

Brocard points

imaginary foci of the Brocard ellipse

A', B', C' : vertices of the ITB triangle

other points below

Geometric properties :

The ITB triangle A'B'C' is defined at K1098 and 𝞱 denotes the isogonal conjugation with respect to this triangle.

K1103 is the locus of M such that the midpoint of M and 𝞱(M) lies on the Brocard axis of ABC which is also the Brocard axis of A'B'C'. It follows that K1103 is an isogonal focal nK with respect to the ITB triangle hence it is K019 for this triangle.

K1103 is the locus of contacts of tangents drawn through X(99) to the circles with center on the line Ω1Ω2 which are orthogonal to the Brocard circle.

K1103 is also an isogonal focal pK with respect to the (yellow) triangle (T) whose vertices are X(99) and the Brocard points Ω1, Ω2 which are the same in both triangles ABC and A'B'C'. K1103 is therefore the locus of M such that the line passing through M and 𝞱(M) is parallel the Brocard axis.

Ω is the circumcenter of (T) and X is the antipode of X(99) on the circumcircle (Ω) of (T). X is the common point of K1103 and its real asymptote which is the line parallel through X(98) to the Brocard axis.

The third point Q of K1103 on the line Ω1, Ω2 is the orthogonal projection of X(99) on this same line.

K1103 and K019 meet again at P1, P2 on the line X(3), X(98), X(98) and on the 2nd Brocard circle (C) with center X(3) passing through the Brocard points.

The polar conic of X(511) in K1103 is a diagonal rectangular hyperbola (H) with respect to ABC. It passes through the in/excenters of both triangles ABC, (T) and also X(39), X(511), X(512), X(3229).

These points P1, P2, X, Q, Ω are now X(32481), X(32482), X(32483), X(32484), X(32485) in ETC (2019-05-13).