Theorem 5

         If ABC a triangle then sin²(A/2) , sin²(B/2), sin²(C/2) are roots of the equation:

               16R² x³ - 8R ( 2R-ρ ) x² + (  τ²  + ρ² -8Rρ ) x -ρ² =0

Theorem 4

                If ABC a triangle ,then  1/sinA, 1/sinB,1/sinC are roots of the equation :

                       2τρ x³ - ( τ ² + ρ² + 4Rρ ) x² +4Rτ x -4R² = 0

Theorem 3

                      If ABC a triangle then cosA , cosB , cosC are roots of the equation :

                                  4R² x³ - 4R ( R+ρ ) x² + ( τ²+ρ²-4R²) x+ (2R+ρ)² - τ² = 0

Theorem 2

                If ABC a triangle then  sinA , sinB, sinC  are  roots of the equation :

                       4R² x³ -4Rτ x² + ( τ² + ρ² +4Rρ ) x -2τρ=0

Theorem 1

             


                 If ABC a triangle with sides of length a,b,c then a,b,c are  roots of the equation :
                       
                                  x³ -2τ x² + ( τ² + ρ² + 4 R ρ ) x -4R ρ = 0

                          τ : semi-perimeter
                          ρ : radius of the inscribed circle 
                         R : radius of the circumscribed circle