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Please refer to the document, "Archimedes' Quadrature of the Parabola":

https://www2.bc.edu/mark-reeder/1103quadparab.pdf

This document describes how Archimedes proves that the area of any parabolic segment is equal to four-thirds the area of the inscribed "vertex triangle".

As a preliminary, on pages 6 and 7 are given Archimedes' three properties of the parabola, that seem to have been well-known enough in Archimedes' time, that he omits the proofs. These properties refer to a parabolic segment, and its vertex triangle:

1) Tangent property

2) Bisecting property

3) Equation of the parabola

In the referenced document, these three properties are proved using the modern techniques of algebra and coordinate geometry. But how would the ancients have proved these properties? I looked through the CONICS of Apollonius, but didn't find these proofs (though I may have missed them).

Any help in finding the ancient proofs much appreciated!

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The analogue of the equation of the parabola is given by Apollonius in I.11 (Proposition 1 in Heath's translation) and restated in I.20. The tangent and bisecting properties are given by Apollonius in I.17 and I.32 (Prop. 11 in Heath). Because, as explained at the beginning of the treatise, a diameter is a line (parallel to the axis of a parabola) bisecting a pencil of parallel chords of a parabola, while any half-chord is called an ordinate to that diameter.

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The synthetic proofs at cuttheknow.org I think are very straightforward: See https://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesTriangle.shtml

I put together a side by side synthetic/analytic comparision as well @ https://mymathclub.blogspot.com/2019/06/quadrature-of-parabola-proposition-2.html

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