The Triangle of Smallest Area Which Circumscribes a Semicircle

TL;DR

This paper solves the problem of finding the smallest-area triangle that circumscribes a semicircle, introduces a sequence of generalized golden right triangles, and presents a construction for the maximum such triangle.

Contribution

It provides a solution to a classical geometric problem and introduces a new sequence of generalized golden right triangles with a specific construction.

Findings

Identified the triangle of smallest area circumscribing a semicircle.

Developed a sequence of generalized golden right triangles.

Constructed the maximum generalized golden right triangle T_2.

Abstract

An interesting problem that determine a triangle of smallest area which circumscribes a semicircle is solved. Then a generalized golden right triangles sequence $T_n$ is obtained, and an interesting construction of the maximum generalized golden right triangle $T_2$ is also shown.