Search Results for “arcs”
Rotating Ellipses with Arcs
Now that you know how two ellipses can rotate such that they are tangent to each other and a third larger ellipse (see this app); the next challenge is to figure out how an arc placed on each ellipse can be constrained (with proportional points along the ellipses) such that each arc always covers half of the ellipse and one endpoint is on the tangent point of the two smaller ellipses.
Archimedes Twins
The large circle is tangent to each of the purple circles, which are tangent to each other. The radius of the large circle is the sum of the radii of the purple circles. The Yellow circles are each tangent to one of the purple circles, to the large circle and to the common tangent of the two purple circles. The yellow circles are called the Archimedes Twins, and have the same radius. Can you work out what this radius is in terms of the radii of the purple circles?
Archimedes Angle Theorem
A is the circle center, and |AB| = |BD|. What is the relationship between angle CAE and angle BDA? You can drag B and C and watch it change. Can you prove the relationship? This is an example from Archimedes Book of Lemmas.
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.
An Arbelos Theorem
The large circle in the diagram has radius 1. Circles centered at A and B are tangent and each tangent to the large circle. (The shape in between the circles is called an arbelos after the Greek for a cobbler's knife, which it apparently resembles.) BC is tangent to circle A and AD tangent to circle B. EF and GH are perpendicular to AB. Can you show that they are equal? Can you find an expression for their length in terms of the radius of circle A? Are they the same size as Archimedes "twins"
Parabola Generalization
A 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.
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.
