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Q.E.D.

28 29 FoldIng conIcs burning mirrors whispering walls Mark a dot on a circular piece of paper, fold a perimeter point onto the dot to produce a crease line, and repeat for various points on the perimeter. An ellipse will begin to appear opposite, top. The proof that this works hinges on the defining property of ellipses in terms of two focal points see previous page and the property that motivates the term focal point If we make the inside of an ellipse into a mirror, every light ray originating at one of its two focal points will pass through the other, having been reflected off the ellipse below, left. This is the principle behind burning mirrors and whispering walls. Put a candle at one of the focal points and its heat will focus at the other, whisper in one of the focal points of a large elliptical wall and your friend in the faraway other focal point will be able to hear you clearly. In general, light rays that miss the focal points envelope ellipses or hyperbol sharing the focal points with the initial ellipse below, middle and right. If you replace the circular piece of paper with a rectangular one and fold from one of its sides only lower, opposite, you get part of a parabola. From this, we can reconstruct the definition of a parabola in terms of a focal point and a line, and see how Archimedes came up with the idea of a parabolic array of mirrors focusing the sun to burn warships.
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