Steiner–Lehmus theorem

The Steiner–Lehmus theorem states that if two internal angle bisectors of a triangle have equal lengths, the opposite sides are equal and the triangle is isosceles.

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Angle bisector: A ray or segment that divides an angle into two equal angles. The theorem compares the lengths of two such segments inside one triangle.
Jakob Steiner: A Swiss mathematician known for work in geometry and for founding the journal Journal für die reine und angewandte Mathematik. Steiner publicly posed the angle-bisector problem that became associated with his name.
Converse of the Steiner–Lehmus theorem: The claim that equal opposite sides in a triangle imply equal lengths of their internal angle bisectors. It reverses the theorem’s implication and follows from the triangle’s symmetry.
Synthetic geometry: Geometry developed through direct reasoning about figures, without coordinates or algebraic representation. Classical proofs seek a direct geometric route from equal bisectors to equal sides.
Angle bisector length formula: A formula expressing an internal angle-bisector length using adjacent sides and the opposite side. Its expression lets equal bisector lengths be translated into relations among side lengths.
Ludwig Lehmus: A German mathematician and professor whose name is attached to the angle-bisector theorem. Lehmus communicated the problem to Steiner, giving rise to the theorem’s joint name.
External angle bisector: A line that divides an exterior angle of a triangle into two equal angles. The theorem concerns internal bisectors; substituting external ones changes the claim.
Trigonometry: The study of relationships between angles and side lengths, especially through trigonometric functions. Sine-rule arguments can compare the two bisectors through the triangle’s angles.
Law of cosines: A triangle identity relating the square of one side to the other sides and the cosine of the included angle. It can connect the triangle’s angles and side lengths in algebraic proofs.
Journal für die reine und angewandte Mathematik: A mathematics journal founded in 1826 by August Leopold Crelle. The early discussion of the problem appeared in the mathematical publishing context surrounding this journal.