Bernoulli's Lemniscate Formed from an Isosceles Trapezoid

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Let

ABCD

be an isosceles trapezoid with horizontal bases.

ABCD

has an inscribed circle

σ

if the altitude of the trapezoid equals the geometric mean of the bases. Let

G

and

H

be the intersections of the circles with diameters

AB

and

DC

. Then the points

G

and

H

lie on a Bernoulli lemniscate.