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