We present an elementary proof of Euclid's Fifth Postulate. The proof uses only the tools of Book I of Euclid's Elements — straight lines, circles, angle bisectors, and the congruence of triangles. No similarity theory, no properties of parallel lines, and nothing beyond what Euclid himself had available is used. The key insight is this: the bisectors of the interior angles formed by a transversal cutting two line segments construct a point P equidistant from all three lines. This point P, established purely by the RHS (Right angle, Hypotenuse, Side) congruence criterion, can exist as a finite interior point only together with a corresponding excenter Q. The existence of P and Q as the incenter and excenter of a triangle formed by the three lines forces AB and CD to meet at a finite point. They meet on the side where the sum of the interior angles is less than two right angles — precisely the content of Euclid's Fifth Postulate. The proof is, to the best of our knowledge, the first proof of Euclid's Fifth Postulate that is free of circularity and proceeds without invoking any principle beyond Book I of Euclid's Elements.
Radhakrishnamurty Padyala (Sat,) studied this question.