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 .
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.
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 .


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[1] D. S. Modic, Triangles, Constructions, Algebraic Solutions (in Slovenian), Ljubljana: Math Publishers, 2009, p. 82.
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