  It is important in drawing techniques to find combinations of two straight
lines and their angle bisectors whose slopes are all rational numbers. This
problem is reduced to solving the Diophantine equation $(a-c)^2(b^2+1) =
(b-c)^2(a^2+1).$ In this article, we describe all non-trivial integral
solutions of the equation with solutions of negative Pell's equations. The
formula is proven by certain properties of solutions of Pell's equations like
those of half-companion Pell numbers and Pell numbers. We also give a formula
for its rational solutions produced by Pythagorean triples with identical legs.
