# 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.

## Solution

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.

## Proof

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.

### Reference

- C. W. Trigg,
*Mathematical Quickies*, Dover, 1985

### Problems (Use a compass only)

- Multiply a Line Segment by a Whole Number
- Reflect a Point in a Line Sgement
- Cross a Circle by a Line Segment
- Drop a Perpendicular to a Line from a Point
- Detect Collinearity
- Complete a Parallelogram
- Bisect an Arc
- Find the Points of Intersection of a Circle with a Line Segment
- Build a Square on a Given Side
- Find the Fourth Proportional of Three Lengths
- Find the Intersection of Two Straight Lines
- Divide a Line Segment into a Whole Number of Parts
- Find the Center of a Given Circle
- Bisect a given line
- Mascheroni Construction of a Regular Pentagon
- A Compass Only Construction - A Chord Tangent to an Inner circle

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