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 a**c**b = v (that is, with apex in point 'c') is known.

Angle a**b**c is less than 90 degrees (so c**a**b 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:

(EDIT: Damn it's hard to get this right, but now I'm positive!)

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 dc**a** = dc**b**, 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

1 Attachment(s)

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

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.

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 a**c**b, then I know that angle a**m**b, where **m** is the center of the circle abc describe, is twice that.

Beautiful!