The See-Saw Lemma

vertical plot range

width

GC	≈	0.43	GD	≈	0.43
CD	≈	0.87	2 1 AB + 1 EF	≈	0.87

AO

≈

1.00

AB×EF

≈

1

OE

≈

1.00

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:

2

1

AB

+

1

EF

,

and (3) AO and OE equal the geometric mean of AB and EF:

AB×EF

.

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