Remember -- use your compass
and straightedge only!
Locate the Center of a Circle
What do you do when a construction problem involving circles, gives you a starting circle such as that shown at the right?
There is NO CENTER indicated on the circle. Unfortunately, you can NOT plot your "best guess" of where you think the center may be located.
If you encounter this situation, you will have to CONSTRUCT the location of the center.
Given: Circle with no center indicated Construct: locate the center of the circle
STEPS: 1. Draw an inscribed angle (an angle with its vertex on the circle and sides terminating on the circle). (This construction also works if you draw two chords instead of the inscribed angle. Drawing the ∠ keeps the chords positioned to more clearly find the center.)
2. Bisect each side of the angle (or chord).
3. The point where the bisectors intersect is the center of the circle.
Proof of Construction: There is a theorem that states, "In a circle, the perpendicular bisector of a chord passes through the center of the circle".
The construction created as the perpendicular bisector of , so the center of the circle must lie on this bisector. Since the same thing is occurring with , the center must also lie on . Since O is the only point on both of these bisectors, O must be the center of the circle.
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as the perpendicular bisector of 
, so the center of the circle must lie on this bisector. Since the same thing is occurring with 
, the center must also lie on 