It looks like Philo’s Line cannot be drawn, 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.
“Look! Up in the sky! It’s a bird! No, it’s a plane! No, it’s another giant step forward in the history of trigonometry!
It comes right on the heels of NordaVinci’s Formula for the Area of a Right Triangle! We have just noticed, ladies and gentlemen, that not only the inscribed circle of a right triangle, but also the escribed circle of the same right triangle, are tangent to the hypotenuse at points that are equidistant from the center of the circumscribed circle of the same right triangle!
That might have been noticed before! But I didn’t notice it before! Now that I have, we also have to realize, that the two tangent points, one on the inscribed circle and one on the escribed circle are also the same distance from the two vertices of the right triangle that constitute the endpoints of the diameter of the circumscribed circle. (see picture)
Then there is the the further amazing fact of NordaVinci’s Formula for the Area of a Right Triangle being applied not only with respect to the tangent point of the inscribed circle, but also to the tangent point of the escribed circle!
Both tangent points serve the same function in that they divide the hypotenuse into two line segments, the product of whose lengths is the area of the right triangle!
Further interesting, and a previously recorded fact, is that the radius of the escribed circle opposite the right angle is equal to the inscribed circle radius plus the hypotenuse, which is none other than the famous semiperimeter of the right triangle, meaning obviously that the product of the inscribed radius and the escribed radius (the one just mentioned) is also equal to the area of the right triangle!
Dumbfounding! Amazing! and Flabbergasting!
