A sufficient condition for reciprocal Jacobi triangles

Abstract

Given a triangleABCletA′B′C′be a Jacobi triangle forABC. When∠BA′C′=∠CA′B′=α′,∠AB′C′=∠CB′A′=β′and∠AC′B′=∠BC′A′=γ′, the triangleABCisa Jacobi triangle forA′B′C′. In this case we say thatABCandA′B′C′are reciprocal Jacobitriangles. In 2015, G.T. Vickers presented a necessary condition for two triangles to be reciprocal,but the question whether that condition was also sufficient remained open. In this work we proveit by using basically trigonometric relations.