Let there be a triangle inscribed in a circle such that the longest side of the triangle forms the diameter of the circle. Show that the triangle is a right triangle.
Let be the diameter of the circle. Construct triangle such that is the longest side of the triangle and a point is placed on the perimeter of the circle such that it forms an inscribed angle that is to the arc of the circle.
Consider the diagram shown.
Let the centre of of the circle be point . This means the lines are radii. This demonstrates there exists two isosceles triangles and . According to the diagram, it can be deduced that if and , then .
Since the sum of the interior angles of a triangle is 180 degrees,
This proves that the triangle is a right triangle.