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Generalizations of a Theorem from Archimedes Liber Assumptorum
The first diagram is from Archimedes' Liber Assumptorum (book of Lemmas). The later diagrams are generalizations to conics.Tags: Archimedes, circle, ellipse, parabola, hyperbola
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A lemma of Archimedes
BF is perpendicular to diameter CD. E is the intersection point of tangents at B and D. Observe that BG = FG. Can you prove it? This is from the Liber Assumptorum (Book of Lemmas) translated from Arabic but attributed to Archimedes.
Ellipse Generalization
A theorem of Archimedes on a circle can be generalized to ellipses as shown here. CE is a diameter of the ellipse. FG is parallel to the tangent at C. H is the intersection of the tangents at F and C. Observe that EH bisects FG.
Hyperbola Generalization
A theorem of Archimedes on a circle can be generalized to a hyperbola as shown here. CE is a diameter of the hyperbola. FG is parallel to the tangent at C. H is the intersection of the tangents at F and C. Observe that EH bisects FG.