The See-Saw Lemma
Draw a semicircle with base AE and center at O. Let the tangent line to the semicircle at D meet the perpendiculars drawn from A and E at B and F. Let G be the point of intersection of BE and AF. Let C be the projection of D onto AE. Then there are three results: (1) GC = GD, (2) CD is equal to the harmonic mean of AB and EF: and (3) AO and OE equal the geometric mean of AB and EF: .[more]
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For more information see See-Saw Lemma: What Is This About?.