Bisect an Arc
Geometric Construction with the Compass Alone
Let two points A and B belong to a circle with center O. Bisect the two arcs of the circle defined by the points A and B.
Start with completing two parallelograms (Problem #6): ABOC (C is opposite to B) and OABD (D is opposite to A). Draw two circles with radius
Note that the arcs AB do not have to be actually drawn.
Let R be the radius of the original circle. By construction, the points C, D and O are collinear (lie
on the same straight line.) The same is true for the points F, G and O. Furthermore, OF is perpendicular
to CD. All we have to show is that
For a parallelogram XYZW we have the following identity
|XZ² + YW² = 2(XY² + YZ²)|
which I'll apply to the parallelogram ABOC to get
A rather specialized problem [Trigg, #248] asks to divide a circle with a known radius into four equal parts. The problem is easily solved by first finding two diametrically opposite points and then applying the above construction to the semicircles so defined. Starting with an arbitrary point A, mark on the circle points C, D, B such that the distances AC, CD, DB all equal the radius of the circle. Point B is opposite A on the circle, so that AB is a diameter.
- C. W. Trigg, Mathematical Quickies, Dover, 1985
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