Central angle intersection points and of the legs of the angle with the circle form a diameter, then is a straight angle. (In radians, .) Let be the minor arc of the circle between points and , and let be the radius of the circle. If the central angle is subtended by , then If the central angle is not subtended by the minor arc , then is a reflex angle and If a tangent at and a tangent at intersect at the exterior point , then denoting the center as , the angles (convex) and are supplementary (sum to 180°). A
Angle trisection and the mark on the line is labeled . This ensures that . A radius is drawn to make it obvious that line segments , , and all have equal length. Now, triangles and are isosceles, thus (by Fact 3 above) each has two equal angles. Hypothesis: Given is a straight line, and , , and all have equal length, Conclusion: angle . Proof: Clearing, , or , and the theorem is proved. Again, this construction stepped outside the framework of allowed constructions by using a marked straightedge. Thomas Hutcheson published an article in the Mathematics Teacher that used a