The Expanding Circle Construction

There can never be enough conic-section construction techniques—at least that’s my philosophy, having grown up to think that conics existed purely in the realm of algebraic equations. So to continue my conic section construction series on Sine of the Times, I’ll present a parabola construction attributed to Ibn Sina (Avicenna), a Persian polymath (c. 970 – 1037) who was the preeminent philosopher and physician of the Islamic Golden Age.

In the Web Sketchpad model below (and here) Points B and C sit on the y-axis with point C as the center of a circle passing through point B. Points E and F mark where the circle intersects the x-axis. Points G and H sit at the intersection points of the tangent to the circle through point D with the lines passing through points E and F that a parallel to the y-axis. As you drag point C along the y-axis, observe the traces of points G and H. Can you prove that they form a parabola?

Page 2 of the websketch allows you to construct this model from scratch using a small set of tools. The video at the end of this post demonstrates the construction steps.

You’ll find this construction, as well as many others, in my book Exploring Conic Sections with The Geometer’s Sketchpad.

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