Geometric Construction with the Compass Alone

Construct the center of a given circle.

Solution

We are given a part of a circle. On the diagram it's the arc whose ends point up. Point G will be shown to be its center. We proceed in several steps:

1. On the arc, choose a point A. With A as a center and an aribtrary radius, draw circle I that intersects the given arc at two points - B and D.
2. Use Problem #1 to construct the point C such that BC forms a diameter of the circle I.
3. With the radius CD draw two circles - one centered at A, another at C, and denote by E the point of their intersection.
4. Draw a circle of radius CD centered at E. This intersects the circle I at the point F.
5. Now, the segment BF is the radius of the given circle whereas the two circles drawn with this radius and centers at B and A intersect at its center.

Proof

The isosceles triangles ACE and AEF are congruent, therefore EAF = ACE. Further, BAE = ACE + AEC for BAE is an exterior angle of the triangle ACE. Also, BAE = BAF + EAF. Which gives BAF = AEC.

Thus, the isosceles triangles ABF and ACE are similar which implies BF/AB = AC/CE or BG/AB = AC/CD. It follows from the latter that the isosceles triangles ABG and ACD are also similar and, hence,

BAG = ACD = BAD/2 = DAG

with the latter two equalities following from the fact that

BAD = ADC + ACD = 2ACD = 2BAG.

Since BAG = DAG, we conclude that the isosceles triangles ABG and ADG are congruent and, therefore, BG = AG = DG. It follows trhat the point G is the required center of the given arc.