DEF is pivotal triangle of ABC around the first Brocard point J of ABC. There is another similar triangle resulting from the other orientation of the triangle and related to the second Brocard point of ABC. Anyway, to see that DEF is pivotal one does an easy calculation of the ratios (|BE|:|CD|:|AF|) and shows that they are equal to ((c/b):(a/c):(b/a)). Thus, applying the well known criteria (see BrocardPivot.html ), DEF is pivotal around the Brocard point J and similar to ABC. J is also the first Brocard point of triangle KLM.

The circles {BEF}, {CED} and {ADF} are tangent to the sides ED at E, DF at D and FE at F respectively. Joining an arbitrary point N of the circle {CED} with the vertices D, E and intersecting with the other two circles, we create external pivotal triangles NOP. Point J is the first Brocard point of all these triangles. Their symmedian point S moves on the Brocard circle of DEF, as N varies on {CED}.

There is a particular position of N, where the sides of NOP are orthogonal to those of DEF. Then S coincides with the circumcenter Q of DEF. The corresponding configuration is then identical to the one discussed in Brocard2.html .

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