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!

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- Jan 16th 2013, 07:00 AMEncircledCircle 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! - Jan 16th 2013, 07:49 AMearbothRe: Circle from two points and an angle
Hej,

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

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

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:

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

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

so $\displaystyle r = \frac{\frac12 |\overline{NS}|}{\sin\left(\frac12 c^\circ\right)}$ - Jan 16th 2013, 08:16 AMEncircledRe: 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. - Jan 16th 2013, 10:07 AMSorobanRe: Circle from two points and an angle
Hello, Encircled!

Quote:

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

Chord $\displaystyle N\!S = d.$

. . Midpoint $\displaystyle M\!:\;MS = \tfrac{d}{2}$

Angle $\displaystyle NV\!S = \theta.$

Draw radii $\displaystyle ON = OS = r.$

$\displaystyle \angle NOS = 2\theta;\;\angle MON = \angle MOS = \theta.$

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

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

- Jan 24th 2013, 11:42 AMEncircledRe: Circle from two points and an angle
I rephrased the question here:

http://mathhelpforum.com/geometry/21...gle-known.html

(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.

So I can't solve for radius according to your formula there.

**Soroban**:

Why would angles NOS = 2 NVS, and MOS = NVS?

It looks wrong. - Jan 25th 2013, 02:22 AMearbothRe: Circle from two points and an angle
That isn't necessary. I wrote you:

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

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

$\displaystyle 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. - Jan 25th 2013, 05:05 AMEncircledRe: 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