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By Larry Ottman
Alternate Hyperbola Construction
This app allows you to experiment with an alternate construction of a hyperbola.  The tracing of the perpendicular bisector of the segment between a point on a circle and a point outside the circle create a hyperbolic locus. (G-GPE 3)

Tags: Geometry, Common-Core, conic-sections, hyperbola, bisector, perpendicular, circle

By Larry Ottman
Alternate Ellipse Construction
This app allows you to experiment with an alternate construction of an ellipse.  The tracing of the perpendicular bisector of the segment between a point on a circle and a point in the circle create an elliptical locus. (G-GPE 3)

Tags: Geometry, Common-Core, conic-sections, ellipse, bisector, perpendicular, circle

By Phil Todd
Euclid Book 6 Proposition 8
If in a right-angled triangle a perpendicular is drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another.

Tags: Euclid

By admin
Circle Involute
In this clock each hand is a piece of string attached to the edge of a circle. The string is pulled tight at the twelve o?clock position and as the hand moves, it wraps around the circle, and thus gets shorter.?The curve is called the circle involute. It is used for designing gear teeth.  As the strings are stretched tight, they are perpendicular to the involute curve and tangential to the circle.

Tags: clock

By admin
Archimedes Trammel
Two points on each hand slide in perpendicular grooves.  The end of the hand traces an ellipse.  This device was invented by Archimedes and is called Archimedes Trammel.  I?ve also heard it called the Cincinnatti do-nothing machine and the Bulls**t Grinder.  Watch the clock, in Cincinnatti or elsewhere, but don't do nothing.  And don't grind, especially your teeth.

Tags: clock

By Phil Todd
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"

Tags: archimedes, arbelos, puzzler

By Phil Todd
Tchirnhausen's Cubic
The caustic formed by light projecting perpendicular to the axis of a parabola is called Tchirnhausen's Cubic. What happens when the light projects at some other angle?

Tags: Parabola, caustic

By Duncan
And the envelope please...
What is the phantom curve you see when you look at a set of lines, perpendicular to a set of chords through a common point in a circle?

Tags: envelope, circle, ellipse, hyperbola

By Phil Todd
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

By Phil Todd
Tangents to Polar Functions
The purple curve has polar equation r=f(θ). A lies on this curve, and B is a point at the intersection of the tangent at A with the line perpendicular to OA. The red curve is the locus of B The grey curve is the curve r=g(θ) rotated by the quantity on the slider. What function g() will rotate to lie on top of the red curve?

Tags: polar-function, tangent, spiral, Archimedes

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