4. Construct a Triangle Given Its Circumradius, Inradius and a Vertex Angle

This Demonstration constructs a triangle given its circumradius , inradius and the angle at .

Construction

Draw a circle with center of radius and a chord of length . Let be the midpoint of and let be an endpoint of the diameter perpendicular to .

Step 1: Draw a line parallel to at distance from intersecting at .

Step 2: Draw a second circle with center through and . Let be one of the points of intersection of and .

Step 3: Let be the intersection of and .

Step 4: Draw the triangle and its incircle.

Verification

Theorem: Let . Then .

Proof: In the right-angled triangle , the hypotenuse length is and the leg , so ). Since the arc equals the arc , bisects . The distance of to is , so is the incenter of .