A Problem
of Wetzel's: Triangle in Rectangle
Lu Yang
cdluyang@mail.sc.cninfo.net
Inst. of Educational Softwares
Guangzhou University (GUI HUA GANG Campus)
Zhenbing Zeng
zeng@mail.sc.cninfo.net
Automated Reasoning
Chengdu Institute of Computer Application
China
Abstract
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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