Given an angle BAC and a point X, interior to the angle, find the shortest line segment through X reaching from ray AC to ray AB. (see figure)Philon's discovery was that the shortest segment occurred when the distance from the point X to one ray was equal to the distance from the other ray to the foot (G in the figure) of the perpendicular to the segment from A .

Philon's only surviving work is his treatise on mechanics. It appears that he travelled throughout the Eastern Mediterranean studying the construction of catapults. He may have been a militrary advisor of some sort. He is mentioned in the writings of Heron of Alexandria. You can find more about his life at the St Andrews University web site.