Brocard2

Brocard points (2)

This is the typical construction of the [first] Brocard point of triangle ABC: Take the orientation A- >B- >C and draw the circles {ABD} tangent at BC, {BCD} tangent to CA and {CAD} tangent to AB. They intersect at a common point D. This is the first Brocard point of ABC (see Brocard.html ). Consider now triangle HIJ, which has as vertices the centers of the previous circles. It is similar to the original triangle ABC and has the same first Brocard point D with ABC. The circumcenter O of ABC is the second Brocard point of HIJ. O is also the symmedian point of the triangle KLM, similar to ABC, whose vertices are the intersection points of the orthogonals MB to BC, KA to AB and JC to AC.

Triangles ABC and HIJ can be considered as particular instances of a triangle SQP, pivoting around the first Brocard point of MKL. All these triangles are similar to each other and to MKL. In fact this is a basic property of the Brocard points: Its [pedal] triangles are similar to the original triangle. Triangle ABC is characterized by having its sides orthogonal to those of MKL. HIJ is characterized by having its vertices at the middles of segments CL, BM and AK. The question of when three points SPQ on the sides of a triangle MKL, define a similar to MKL triangle, pivoting around its first Brocard point, as well as another particular instance of a pivoting triangle is discussed in the file BrocardPivot.html .

In the figure above the line OD'' is the locus of the second Brocard points D' of the pivotal triangles. It passes through the symmedian point O (and circumcenter of ABC) of MKL and its second Brocard point D''. The discussion on pivots is initialized in Pivot.html .

http://users.math.uoc.gr/~pamfilos/