Hello friends, I want to present you a work on the Phi-Pi triangles, their properties and interrelation between the number Phi and the number Pi within the circumference.
In the first place we see what the Phi triangle consists of.
As we can see in the drawing, in the right triangles in addition to the interrelation between the sides by Pythagoras Theorem, there is also a special triangle whose sides are also interrelated by the number Phi.
In this case, the legs of this triangle are related to the hypotenuse (h) by the number Phi, the minor leg being a = h/phi and the bigger leg being b = h/Sqt Phi.

Well, by finding the square root of the segment Pi/2, we obtain the semi-diagonal necessary to construct the square in the quadrature of the circle, as we also see in the drawing.

Finally, we remember how the segments Phi and the square root of Phi are obtained, necessary to integrate them in the drawing of the circumference, and to be able to square the circle in an approximate way with a compass and ruler only.