HomothetyProperty

Consider three lines emanating from point C: CA, CB, CD. Draw the lines BD and AD. Take then points F, E, moving respectively on CA, CB, such that the ratios (CE/CB) = (CF/CA) = k. Then the parallels from F and E to AD and BD, respectively, meet on line CD.

The converse is also valid. For points A, B, C, D as before, defining the five lines CA, CB, CD, AD, BD and a moving point I on CD, project I parallel to AD, BD respectively, on line CA to point F and on line CB to point E. Then (CE/CB) = (CF/CA) . For an application of this remark to an interesting question on polygons, look at SuccessiveArcsHex2.html .

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