Abstract Presented at the 10th Asian Technology Conference in Mathematics
December 12-19, 2005, South Korea

A Problem of Wetzel's: Triangle in Rectangle

Lu Yang
Inst. of Educational Softwares
Guangzhou University (GUI HUA GANG Campus)

Zhenbing Zeng
Automated Reasoning
Chengdu Institute of Computer Application


In this paper, we present a symbolic solution to an open problem of J.E.Wetzel: find the necessary and sufficient condition for a triangle of sides a, b, c to fit into a rectangle of sides p,q. Our method is first transform it into a quantifier ellimination problem through analyzing on the maximal fitting configurations and then solve the QE problem by using of symbolic computation tool MAPLE. The result is: A triangle of sides a,b,c can be fitted into a rectangle of sides p,q if and only if the following at least one of the following three systems of inequalities holds:

S1: a^2*q<=2*p*S<=a*p*q and b^2<=c^2+a^2 and c^2<=a^2+b^2;
S2: a^2<=p^2+q^2 and 4*p*S<=(a^2+b^2-c^2)*q and 4*q*S<=(c^2+a^2-b^2)*p;
S3: 4*b^2*p^2+4*c^2*q^2-16*p*q*S>=(b^2+c^2-a^2)^2 and (a^2+b^2-c^2)*S<4*q*S<2*b^2*p and (c^2+a^2-b^2)*S<4*p*S<2*c^2*q.

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