Equation for the sidelengths of a triangle in terms of its three angle bisectors

We give a 20th degree polynomial one of whose roots is one of the sidelengths of the triangle.

Happy New Year!

 threebisectors.pdf

Euclid 3639 - Some loci of circumcenters

We study a problem proposed by Antreas Hatzipolakis at Euclid 3639 

PDF: Euclid 3639

A circle with huge equation

 Antreas Hatzipolakis proposes the problem Euclid 3637 as follows:

Let ABC be a triangle, L the Euler line, P a point on L and A'B'C' the pedal triangle of P
Denote
A", B", C" = the orthogonal projections of A, B, C, on L, resp.
(Oa), (Ob), (Oc) = the circles with diameters A'A", B'B", C'C", resp.
The radical center of (Oa), (Ob), (Oc) lies on the NPC.
Which point is it in terms of P?

Here is the Mathematica code and its output to the solution of this problem and its generalization for any line $px + qy + rz =0$.

Solution to Euclid Problem 3637

A figure for Problem Euclid 3630

A figure for Problem 3630