center of a circle: prove construction is correct

[soccerisgreat]

I've done a construction to determine the center of a circle. I used 2 chords. Found their perpendicular bisectors and the intersection is the center of the circle, but how do I prove that it is the center?

 

[Deleted member 4993]

Re: center of a circle

I've done a construction to determine the center of a circle. I used 2 chords. Found their perpendicular bisectors and the intersection is the center of the circle, but how do I prove that it is the center?

You can show that the end points of each chord is equi-distant from the point of intersection of those two perpendicular bisectors. It will be easy to prove that if the two chords have one common end point.

Also you can assert that any chord has only one perpendicular-bisector - and that bisector must pass through center. Thus those two bisectors must intersect at the center.

 

[pka]

Re: center of a circle

I've done a construction to determine the center of a circle. I used 2 chords. Found their perpendicular bisectors and the intersection is the center of the circle, but how do I prove that it is the center?

That is a difficult question to answer. Because the concept of a proof depends upon the axiom set that you are given. We don’t know what that set is.
Usually a circle is defined as a set of points in a plane each of which is equally distant from a fixed point, the center.. Then there is an axiom or a theorem that states three non-collinear points determine a unique circle.

What you have found is a point equally distant from three non-collinear points. Using the axiom/theorem that point is by definition the center.

 

[soccerisgreat]

Thank you! Proving the center point I've found is equidistant from the endpoints of the chord should do the trick.