In this article, we study the property of points X on a line l for which the ratio \(\text{BX}\over \text{CX}\) reaches its maximum and minimum values for two given points B and C. It is known that, in a triangle ABC (AB > AC), the points X on the line l, containing the internal bisector of angle A, where \(\text{BX}\over \text{CX}\) attains its maximum and minimum values, are the incenter and excenter of the triangle, respectively. While known proofs rely on calculus or trigonometry, we present an elementary geometric solution for the general case of the location of line l relative to segment BC, using the Apollonius circle.
