It looks like Philo’s Line can be drawn as below, using only a straightedge and a compass, by virtue of knowing NordaVinci’s Formula for the Area of a Right Triangle, which can be used to show that the two tangent points 1) the inscribed circle tangent point to the hypotenuse, and 2) the escribed circle tangent point to the hypotenuse, are equidistant from the endpoints of the hypotenuse on each end.
NordaVinci 
Some history . . .
http://mathshistory.st-andrews.ac.uk/HistTopics/Doubling_the_cube.html
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Some more history . . .
http://mathshistory.st-andrews.ac.uk/Biographies/Philon.html
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PHD Thesis on IMPOSSIBILITY RESULTS: FROM GEOMETRY TO
ANALYSIS by Davide Crippa
Davide Crippa. IMPOSSIBILITY RESULTS: FROM GEOMETRY TO ANALYSIS : A study in early modern conceptions of impossibility. History, Philosophy and Sociology of Sciences. Univeristé Paris, Diderot Paris 7, 2014. English. fftel-01098493
HAL Id: tel-01098493
https://hal.archives-ouvertes.fr/tel-01098493
Submitted on 25 Dec 2014
Click to access phdNovember.pdf
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