Danie Brink

PEDAL TRIANGLES CONTINUED

 If the point P is on one of the sides of triangle ABC, the point forms one of the vertices of the pedal triangle. The pedal triangle formed with P on one of the sides of triangle ABC will always be on the interior of triangle ABC.

 If the point P is moved to a vertex of the original triangle ABC, the pedal triangle "collapses" to just the altitude of the triangle from that vertex.

 In certain cases the pedal triangle reduces/collapses to three collinear points. The location of P for this to happen is when P is on the circumcircle as indicated alongside. This line is called the Simson Line.

 If we trace the Simson Line as the point P moves around the circumcircle, the envelope formed is a deltoid. The area of the deltoid is half the area of the circumcircle.

 There are three places where the Simson line forms one of the sides of the original triangle. If P is at point K the Simson Line is side AB, if P is at point L the Simson Line is side BC and if P is at point M the Simson Line is side AC.

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