“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!

1 thought on “Dumbfounding! Amazing! and Flabbergasting!”

The Philon (or Philo) Line
“Well begun is half done.” – Aristotle

Philon of Byzantium (circa 100 A.D.) worked on trying to duplicate the cube (construct a cube with volume equal to twice the volume of a given cube) using only a straight edge and compass. Since he could not find a direct solution he kept searching for problems that were equivalent in the sense that one could construct the solution to one problem from the solution to the other by means of the straight edge and compass. Constructing the Philon line (as it is now known) is one such equivalent problem that he discovered. He could not find a solution to the Philon line problem either. During the past 2000 years many of the giants of computing science, including Newton, tried to solve Philon’s problem with straight edge and compass without success. It was only one hundred years ago, using algebra, that this was shown to be impossible.

The Philon (or Philo) Line

“Well begun is half done.” – Aristotle

Philon of Byzantium (circa 100 A.D.) worked on trying to duplicate the cube (construct a cube with volume equal to twice the volume of a given cube) using only a straight edge and compass. Since he could not find a direct solution he kept searching for problems that were equivalent in the sense that one could construct the solution to one problem from the solution to the other by means of the straight edge and compass. Constructing the Philon line (as it is now known) is one such equivalent problem that he discovered. He could not find a solution to the Philon line problem either. During the past 2000 years many of the giants of computing science, including Newton, tried to solve Philon’s problem with straight edge and compass without success. It was only one hundred years ago, using algebra, that this was shown to be impossible.

From http://cgm.cs.mcgill.ca/~godfried/research/philon.html

https://en.wikipedia.org/wiki/Godfried_Toussaint (Recently deceased July 2019)