Trigonometry/Circles and Triangles/Brocard's Theorem

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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 = ω
  • BE where E is between A and C, and angle EBC = ω
  • CF where F is between A and B, and angle FCA = ω

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

cot(ω) = cot(A) + cot(B) + cot(C).

From symmetry, there is a second Brocard point, using the same angle ω, at the intersection of the three lines

  • AD' where D' is between B and C, and angle D'AC = ω
  • BE' where E' is between A and C, and angle E'BA = ω
  • CF' where F' is between A and B, and angle F'CB = ω