Some Remarks on $\tau$-Congruent Numbers

In this paper, we extend the work of \cite{Chahal} in several directions. We first determine all Heron triangles that tightly circumscribe the unit circle and the associated $\tau$-congruent numbers generated by them. We then characterize all rational right triangles that tightly circumscribe the unit ellipse and identify the corresponding congruent numbers. In addition, we study of the congruent numbers from the excircle opposite a vertex of a rational right triangle, that is, the circle tangent to one side of the triangle and to the extensions of the remaining two sides.