# Thread: Circle from two points and an angle

1. ## Circle from two points and an angle

I want to determine the circle through points N, S and V.
I know the distance between N and S.
I know the angle NVS.

How could an expression describing such a circle be constrained given that information?
Thank you!

2. ## Re: Circle from two points and an angle

Originally Posted by Encircled
I want to determine the circle through points N, S and V.
I know the distance between N and S.
I know the angle NVS.

How could an expression describing such a circle be constrained given that information?
Thank you!
Hej,

if - and only if - the 3 points are placed as in the attached sketch then

$\angle(NVS) = (x+y)^\circ$

The red lines are the radii which form 2 isosceles triangles which can be split into right triangles.

The angle c at the center is calculated by:

$(180^\circ-2x^\circ) + (180^\circ-2y^\circ) = c^\circ$

The triangle $\Delta(NSC)$ is an isosceles triangle.

so $r = \frac{\frac12 |\overline{NS}|}{\sin\left(\frac12 c^\circ\right)}$

3. ## Re: Circle from two points and an angle

Great answer, and fast, thanks alot!
I can puzzle together the rest of what I need from this point. Magnificent.

4. ## Re: Circle from two points and an angle

Hello, Encircled!

I want to determine the circle through points N, S and V.
I know the distance between N and S.
I know the angle NVS.

How could an expression describing such a circle be constrained given that information?
Code:
              * * *
*           *
*  d/2  M  d/2  *
N o- - - - + - - - -o S
\ *     |     * /
*  \   * θ|θ *r  /  *
*   \     o     /   *
*    \    O    /    *
\       /
*     \     /     *
*     \   /     *
*    \θ/    *
* o *
V
We have a circle with center $O.$
Chord $N\!S = d.$
. . Midpoint $M\!:\;MS = \tfrac{d}{2}$
Angle $NV\!S = \theta.$

Draw radii $ON = OS = r.$
$\angle NOS = 2\theta;\;\angle MON = \angle MOS = \theta.$

In $\Delta SMO\!:\;\sin\theta \,=\,\dfrac{\frac{d}{2}}{r} \:=\:\frac{d}{2r}$

Therefore: . $r \:=\:\frac{d}{2\sin\theta}$

5. ## Re: Circle from two points and an angle

I rephrased the question here:
Want radius of circle through 3 points. A chord and an angle are known.
(Because at first I actually failed to find this, my own, old thread, duh)
Please see it in order to maybe clearer understand what I'm looking for.

And actually, now when I look at it, there are some misunderstandings in this thread here.

earboth:
I do not know angles x or y. I only know the angle NVS = x+y.

Soroban:
Why would angles NOS = 2 NVS, and MOS = NVS?
It looks wrong.

6. ## Re: Circle from two points and an angle

Originally Posted by Encircled
...

And actually, now when I look at it, there are some misunderstandings in this thread here.

earboth:
I do not know angles x or y. I only know the angle NVS = x+y.

...
That isn't necessary. I wrote you:

$(180^\circ-2x^\circ) + (180^\circ-2y^\circ) = c^\circ$

$360^\circ-2x^\circ-2y^\circ = c^\circ$

$360^\circ-2\underbrace{(x^\circ+y^\circ)}_{ \angle(NVS)}=c^\circ$ And now you don't need x and y anymore because you only uses the sum of them and this sum is known.

7. ## Re: Circle from two points and an angle

I borrow this fine ASCII art and introduce in it the point L.
Now, since L is on the opposite side of the chord, as seen from the circle origo O, does it still hold that the angle NLS = half the angle NOS?
It is supposed to hold for any point V on the same circle which the chord crosses, but intuitively, that angle seems much larger when V is at L on the smaller side of a chord. Is another relationship at play there?

Code:
              *L *
*           *
*  d/2  M  d/2  *
N o- - - - + - - - -o S
\ *     |     * /
*  \   * θ|θ *r  /  *
*   \     o     /   *
*    \    O    /    *
\       /
*     \     /     *
*     \   /     *
*    \θ/    *
* o *
V