Area of a triangle and angle bisectors

TL;DR

This paper proves that, given the lengths of a triangle's angle bisectors, it is generally impossible to construct a square with the same area using classical tools or to express the area in radicals of these bisectors.

Contribution

It establishes the fundamental limitations of geometric constructions and algebraic expressions related to triangle area from angle bisectors.

Findings

Constructing a square of equal area from bisectors is generally impossible.

Expressing the area in radicals of bisectors is impossible.

The problem highlights inherent algebraic and geometric constraints.

Abstract

Consider a triangle $ABC$ with given lengths $l_a,l_b,l_c$ of its internal angle bisectors. We prove that in general, it is impossible to construct a square of the same area as $ABC$ using a ruler and compass. Moreover, it is impossible to express the area of $ABC$ in radicals of $l_a,l_b,l_c$.