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

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 $(a-c{)}^2(b^2+1)=(b-c{)}^2(a^2+1)\mathrm{.}$ In this article, we describe all nontrivial 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.