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It is important in drawing techniques to find combinations of two straightlines and their angle bisectors whose slopes are all rational numbers. Thisproblem 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 integralsolutions of the equation with solutions of negative Pell's equations. Theformula is proven by certain properties of solutions of Pell's equations likethose of half-companion Pell numbers and Pell numbers. We also give a formulafor its rational solutions produced by Pythagorean triples with identical legs.

Diophantine equation related to angle bisectors and solutions of Pell's equations