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Simson line and envelope

You can move freely the three points A, B, and C by mouse. The point P moves on the circle as theparameter θ varies.
The points D, E, and F are the foots of perpendicular lines. The line passing through these three points is the Simson Line.
When the point P move on the circumcircle of the triangle, the Simson line draws an hypotrochoid.