Rational Isogonal Conjugates

Given a triangle , draw lines from the vertices to an arbitrary point . Reflect those three lines in the angle bisectors (shown in red). The three new lines intersect in a point called the isogonal conjugate of .

If a triangle has three rational sides, it is called a basic rational triangle. A point is called a rational point of the rational triangle if the distance from to the vertices of is also rational.

It is a theorem that is a rational point of if and only if is a rational point of .

In this Demonstration, rational values and are used to generate a pair of rational isogonal conjugates and in a rational triangle .