The Philo line is a line segment defined from an angle and a point inside the angle as the shortest line segment through the point that has its endpoints on the two sides of the angle. There are some properties of this line in Wikipedia. But I didn't find any information concerning possibility of construction of such line with compass and straightedge. Is it possible and how?
2026-03-30 23:19:45.1774912785
Philo's line construction
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1
No this is not possible:
From the wikipedia article you mentioned, we find that if the philo line is constructible, then we can "double the cube" (i.e. construct a line segment of size $\sqrt[3]{2}$).
From the "doubling the cube" article, we find that this is not possible with compass and straightedge.
Therefore, the Philo line cannot be constructed in general with compass and straightedge.