Circle from two points and an angle

• Jan 16th 2013, 07:00 AM
Encircled
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!
• Jan 16th 2013, 07:49 AM
earboth
Re: Circle from two points and an angle
Quote:

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

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

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 AM
Encircled
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.
• Jan 16th 2013, 10:07 AM
Soroban
Re: 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

We have a circle with center $\displaystyle O.$
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 AM
Encircled
Re: 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.

Soroban:
Why would angles NOS = 2 NVS, and MOS = NVS?
It looks wrong.
• Jan 25th 2013, 02:22 AM
earboth
Re: Circle from two points and an angle
Quote:

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:

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

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

$\displaystyle 360^\circ-2\underbrace{(x^\circ+y^\circ)}_{ \angle(NVS)}=c^\circ$ 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 AM
Encircled
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