Consider a circle, a chord CD of it whose middle is G. Draw two other chords through G: FH and EI. Then lines FI, EH intersect chord CD at points J, K respectively, which are symmetric with respect to G.
The theorem is a consequence of the fact that line LM is the polar of G with respect to the circle. Then lines LF, LE, LG and LM build a harmonic bundle and every line intersecting these lines is divided harmonically by them. Thus, JK being parallel to LM is bisected by LG. Notice that LG, MG are respectively the polars of M and L. The relevant properties of the polars are discussed in the file CyclicProjective.html .
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