Finding the Symbolic Solution of a Geometric Problem Through Numerical Computations

In this paper we prove that if $$L$$ L is the maximal perimeter of triangles inscribed in an ellipse with $$a,b$$ a , b as semi-axes, then $$ (a^2-b^2)^2\cdot L^4-8(2a^2-b^2)(2b^2-a^2)(a^2+b^2)\cdot L^2-432a^4b^4=0 $$ ( a 2 - b 2 ) 2 · L 4 - 8 ( 2 a 2 - b 2 ) ( 2 b 2 - a 2 ) ( a 2 + b 2 ) · L 2 - 432 a 4 b 4 = 0 by accomplishing the following tasks through numeric computations: (1) compute the determinants of matrices of order from $$25$$ 25 to $$34$$ 34 whose entries are polynomials of degree up to $$44$$ 44 , (2) construct a series of rectangles $$R_1,R_2,\ldots ,R_N$$ R 1 , R 2 , … , R N so that if $$L,a,b$$ L , a , b satisfies the relation $$f(L,a,b)=0$$ f ( L , a , b ) = 0 then $$ C_1:=\{(b,L)|f(L,1,b)=0, 0\le b\le 1\}\subset R_1\cup R_2\cup \cdots \cup R_N, $$ C 1 : = { ( b , L ) | f ( L , 1 , b ) = 0 , 0 ≤ b ≤ 1 } ⊂ R 1 ∪ R 2 ∪ ⋯ ∪ R N , and, (3) present a mechanical procedure to decide the validity of $$ R\cap C(F)=\emptyset , $$ R ∩ C ( F ) = ∅ , where $$R$$ R is a closed rectangle region and $$C(F)$$ C ( F ) is an algebraic curve defined by $$F(x,y)=0$$ F ( x , y ) = 0 .