Another Short Proof of the Euler–Chapple Inequality

Choose a triangle and consider the unique circle touching all three sides in the interior.The inradius of the triangle is the radius of this inscribed circle, or incircle.The circumradius is the radius of the circumscribed circle, or circumcircle, which is the unique circle passing through the vertices of the triangle.The Euler-Chapple inequality is a well-known result in geometry that says the following. Euler-Chapple Inequality.Let ABC be a triangle with inradius r and circumradius R. Then the inequality R ≥ 2r holds, with equality if and only if the triangle is equilateral.