21. Construct a Triangle with Equal Base Angles That Are Each Double the Third Angle

This Demonstration constructs an isosceles triangle with base angles that are double the third angle [2].

Construction

1. Draw the line segment .

2. Draw the circle with center and radius .

3. Draw the perpendicular to at with as one point of intersection with .

4. Let be the midpoint of .

5. Let be on the other side of from such that .

6. Let be on such that .

7. Let be on such that .

8. Then meets the required conditions.

Using Euclid II.11 [3], is constructed on such that . The point is constructed such that . Since , is tangent to the circumcircle of (Euclid, III, 37 [4]). Hence, . The exterior angle theorem applied to gives . So . Hence, is isosceles and . So, . Then . Therefore, , .

Euclid IV, 10: "To construct an isosceles triangle having each of the angles at the base double the remaining one." [2].

Euclid II, 11 states: "To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment." [3].

Euclid, III, 37 states: "If a point is taken outside a circle and from the point there fall on the circle two straight lines, if one of them cuts the circle, and the other falls on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the straight line which falls on the circle, then the straight line which falls on it touches the circle." [4].

References

[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998.