# Find the Center of a Given Circle

### Geometric Construction with Compass Only

#### What Is This About?

### Problem

Construct the center of a given circle.

## Solution

We are given a circle. Point G will be shown to be its center. We proceed in several steps:

- 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.
- Use Problem #1 to construct the point C such that BC forms a diameter of the circle I.
- With the radius CD draw two circles - one centered at A, another at C, and denote by E the point of their intersection.
- Draw a circle of radius CD centered at E. This intersects the circle I at point F.
- 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

Thus, the isosceles triangles ABF and ACE are similar which implies

∠BAG = ∠ACD = ∠BAD/2 = ∠DAG |

with the latter two equalities following from the fact that

∠BAD = ∠ADC + ∠ACD = 2∠ACD = 2∠BAG. |

Since ∠BAG = ∠DAG, we conclude that the isosceles triangles ABG and ADG are congruent and, therefore,

### 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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