Finding the Symbolic Solution of a Geometric Problem Through Numerical Computations

Abstract

In this paper we prove that if $L$ is the maximal perimeter of triangles inscribed in an ellipse with $a,b$ as semi-axes, then

$$ (a^2-b^2)^2\cdot L^4-8(2a^2-b^2)(2b^2-a^2)(a^2+b^2)\cdot L^2-432a^4b^4=0 $$

by accomplishing the following tasks through numeric computations: (1) compute the determinants of matrices of order from $25$ to $34$ whose entries are polynomials of degree up to $44$, (2) construct a series of rectangles $R_1,R_2,\ldots ,R_N$ so that if $L,a,b$ satisfies the relation $f(L,a,b)=0$ then

$$ C_1:=\{(b,L)|f(L,1,b)=0, 0\le b\le 1\}\subset R_1\cup R_2\cup \cdots \cup R_N, $$

and, (3) present a mechanical procedure to decide the validity of

$$ R\cap C(F)=\emptyset , $$

where $R$ is a closed rectangle region and $C(F)$ is an algebraic curve defined by $F(x,y)=0$.

This work is supported by the Project No. 20110076110010 from the Ministry of Education of the People’s Republic of China.