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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
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.

Tags: Archimedes, circle, tangent

Ellipse Generalization
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.

Tags: ellipse, archimedes, diameter

Hyperbola Generalization
Hyperbola Generalization
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.

Tags: hyperbola, Archimedes

Parabola Generalization
Parabola Generalization
theorem of Archimedes on a circle can be generalized to parabolas as shown here. Vertical lines correspond to diameters of the parabola. FG is parallel to the tangent at C.  E is the intersection of the tangents at F and C. Observe that teh vertical line through E bisects FG.

Tags: parabola, Archimedes


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