Feynman's lost lecture The motion of the planets around the Sun

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Construct the perpendicular to the tangent passing through F' and then the point G' such that F'T = G'T. Then join G' to P.

Triangles TPF' (green) and TPG' (yellow) are congruent (because TF'=TG' by construction, angle PTF'=angle PTG' and TP common to both sides). Therefore F'P=G'P and angle F'PT = angle TPG'. In the diagram below, the angles double-marked must be equal (as they are "opposite angles"). So the two angles that FP and F'P make with the tangent line must be equal. The distance FG' must equal the length of the string. (Note: this does not seem quite right to me as there seems to be an assumption here that FG' is a straight line.)

These facts mean there is another way of constructing an ellipse which turns out to be important to Feynman's study of planetary motion.