# Want radius of circle through 3 points. A chord and an angle are known.

• Jan 24th 2013, 11:19 AM
Encircled
Want radius of circle through 3 points. A chord and an angle are known.
The distance between points a and b is defined to be 1 length unit.
The angle acb = v (that is, with apex in point 'c') is known.
Angle abc is less than 90 degrees (so cab is more than 90 degrees).

What is the radius of the circle which passes through points a, b and c?

That is, the line 'a' to 'b' is a chord of the circle.
'c' is a third point on the circle which these points describe.

Can the radius 'r' be determined, or at least resticted?

Something like this:
Code:

c     a  o  b
(EDIT: Damn it's hard to get this right, but now I'm positive!)
• Jan 24th 2013, 12:10 PM
Encircled
Re: Want radius of circle through 3 points. A chord and an angle are known.
What about using a fourth point 'd' on the circle and below the middle point 'o' between a and b, such that angle dca = dcb, and such that the circle with center at 'm' would have a radius r = ma = mb = mc = md.

Together with known angle acb and knowing that angle dca = dcb. Isn't that somehow enough info to get r in relationship to the distance of the chord ab?

But note that the points 'c', 'o' and 'd' are not on a straight line. 'd' is instead defined from angles and radius and on a straight line with 'o' and circle center point 'm'
Code:

        m (m is circle center) c     a  o  b         d
• Jan 24th 2013, 02:07 PM
johng
Re: Want radius of circle through 3 points. A chord and an angle are known.
The answer is $\displaystyle r={1\over 2sin(v)}$. Here's a proof:
Attachment 26693
• Jan 24th 2013, 02:22 PM
BobP
Re: Want radius of circle through 3 points. A chord and an angle are known.
You need to make use of two ' circle theorems '.

First is : ' Angles on the same arc are equal '.

What that means is this.
Suppose that A and B are two fixed points on a circle, and that C is a third moveable point on the circle. Then no matter where the point C moves to, so long as it keeps to the same side (arc) of the circle, the angle ACB remains a constant.

Second is : ' The angle at the centre is twice the angle at the circumference '.

Suppose that A, B and C are as described above and that O is the centre of the circle, then angle AOB is double angle ACB.

Apply one after the other and you get the answer to your question.
• Jan 24th 2013, 03:24 PM
Encircled
Re: Want radius of circle through 3 points. A chord and an angle are known.
Yes yes!
I've heard of that before, now it comes back to me. It is not intuitive, rather unbelievable, but undeniably logical.
If I know angle acb, then I know that angle amb, where m is the center of the circle abc describe, is twice that.
Beautiful!