Trigonometry/Circles and Triangles/The Simson line

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Take any triangle ABC and any point P on its circumcircle. Let the perpendiculars from P to AB, BC and CA be D, E and F respectively. (At least one side will need to be produced to meet the perpendicular.) Then D, E and F all lie on a straight line, the Simson line of that point.

This theorem is due to Robert Simson, 1687-1768.