Pedal Triangles

The loci of the midpoints of the sides of pedal triangles form ellipses as the pedal point moves around a circle.    

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Simpson Line and 9 point Circle

The envelope of the Simpson line is seen to be tangent to the 9 point circle and to touch the concentric circle of 3 times the radius.

Tags: pedal-triangle, Simpson-Line, 9-point-circle, circumcircle

Simpson Line

Point E lies on the circumcircle of triangle ABC.  The projections of E onto the sides of the triangle are colinear.  The envelope of such lines forms a deltoid curve. Try dragging E round the circle. Try dragging A B and C.  Note that the envelope curve appears to remain equilateral.

Tags: deltoid, pedal-triangle, circumcircle

Pascal's Limacon

Given a circle with center A and a point C, here are three related ways of constructing Pascal’s Limacon: It is the isotomic (look it up) of the circle with pole C. It is the conchoid (look it up) of a circle centered at A whose circumference passes through C. It is an epitrochoid (look it up) formed by a circle of equal radius rolling around the original. This app constitutes a visual proof of the above, and depends on the fact that the composition of reflections in two parallel lines is equivalent to a translation of twice the distance between the lines.

Tags: Pascal, Limacon, Epitrochoid, Conchoid, Isotomic, Pedal, reflection