Trigonometry/Circles and Triangles/Brocard's Theorem

Brocard's Theorem is due to French mathematician Henri Brocard (1845 – 1922).

[Needs diagram]

Let ABC be any triangle. Draw three lines:
 * AD where D is between B and C, and angle DAB = &omega;
 * BE where E is between A and C, and angle EBC = &omega;
 * CF where F is between A and B, and angle FCA = &omega;

Then the lines AD, BE, CF are concurrent, meeting at a Brocard point, if and only if


 * cot(&omega;) = cot(A) + cot(B) + cot(C).

From symmetry, there is a second Brocard point, using the same angle &omega;, at the intersection of the three lines
 * AD' where D' is between B and C, and angle D'AC = &omega;
 * BE' where E' is between A and C, and angle E'BA = &omega;
 * CF' where F' is between A and B, and angle F'CB = &omega;