On the Compass and Straightedge Construction of Steiner Inellipse Foci and Axes: Synthetic Cevian Theorems on the Curve

This paper addresses the problem of reconstructing a triangle from the points at which its medians intersect its circumcircle. We show that this problem is equivalent to determining the foci of the Steiner inellipse. A new synthetic straightedge-and-compass construction of these foci is developed, based on two classical propositions of J. Neuberg (1889) and on the geometric relationship between the Steiner angles and the Brocard angle. We further introduce an original Euclidean method for determining the direction and orientation of the major and minor axes of the Steiner inellipse. Finally, we establish and prove a new theorem concerning the ratios of the segments determined by a variable cevian at its intersections with the axes of the ellipse. In the particular case of the altitude, these ratios exhibit a cyclic symmetry, and their sum is shown to reduce to the constant (4/3)cot2ω, where ω denotes the Brocard angle of the original triangle.