Consider the acute-angled cone with vertex O and a plane intersecting a generating line OG at a right angle at point A. The plane intersects the cone in the oxytome with diameter AB.
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Consider an arbitrary ordinate (i.e. y value) TK constructed on the axis at T. We wish to determine the relationship between TK and AT, that is, the symptom of the conic. The ordinate TK is located in a horizontal plane that cuts the cone in the circle with diameter DG. In this horizontal plane construct the segments GK and DK, which results in a right triangle inscribed in a semicircle. (The triangle is right by Elements, Book III, Proposition 20). We also know that triangles GTK and KTD are similar (by Book VI, Prop. 8) and this implies
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(1) |
Now consider the similar triangles TAG and TDH in the plane through O, G, and D, the axial plane. The triangles are similar because they each have a right angle and opposite vertical angles. This in turn implies
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(2) |
Also in the axial plane are the pairs of similar triangles HDT and IEA, and BDT and BEA. From these we see that
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(3) |
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(4) |
Combining (3) and (4) we have
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(5) |
Notice also that in triangle IEA the line OL bisects AE so it must also bisect AI, making IA = 2AL. Putting this together with (1) and (2) we have
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(5) |
This might not look like an equation that we recognize, but if we let KT = y, the distance from the center of the ellipse to T be x, AB = 2a, and 2AL = p we have
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This does look like the equation of an ellipse.