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 $\displaystyle r = \frac{\frac12 |\overline{NS}|}{\sin\left(\frac12 c^\circ\right)}$
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?We have a circle with center $\displaystyle O.$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: .$\displaystyle r \:=\:\frac{d}{2\sin\theta}$
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.
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.
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.
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