Geometry, Intersection of Spheres Can someone explain why the intersection of the unit sphere centred at (0,0,0) an the unit sphere centred at (1,0,0) is a circle of radius $\frac{\sqrt3}{2}$ in the plane {$x_1$=1/2}, centred at (1/2,0,0)?
 A: You can see it geometrically:
Just draw unit circles centered at $(0,0)$ and $(1,0)$ in the $(x_1,x_2)$-plane (they intersect in the points $(x_1,x_2)=(1/2,\pm \sqrt{3}/2)$); then rotate the whole picture around the $x_1$ axis.
A: You can also do it algebraically, since both spheres have an equation and the intersection is "solving"/rewriting the equation system (1,2) : 
$$\begin{array}{rr}(1)&x^2+y^2+z^2 = 1\\(2)&(x-1)^2+y^2+z^2 = 1\\\\ [(1)-(2)]: &2x-1=0\\ [((3)+1)/2] :  & x =1/2\\ [(4) in (1)] : & 1/4 + y^2+z^2 = 1\\ [(5)-1/4]:&y^2+z^2 = 3/4\end{array}$$
Now if we gather (4) and (6):
$$\begin{cases} x&=1/2\\y^2+z^2 &= 3/4\end{cases}$$
Which is easier to see that the first one is a plane and the second a circle.
A: The intersection of any two spheres of any size 
that actually do intersect at more than one point is always a circle.
If the spheres are the same size, they are mirror images of each other
through a plane halfway between the two centers of the spheres,
perpendicular to the line through the centers of the spheres.
Moreover, the intersection of the two spheres is a circle in that plane.
Since your two spheres are on the $x$-axis at the points where
$x=0$ and where $x=1$, the "mirror" plane is perpendicular to
the $x$-axis and intersects the $x$-axis at $x=1/2$,
ergo $x=1/2$ is the equation of that plane
and the intersection of the two spheres is a circle in that plane.
All that remains is to find the radius of the circle. Consider any
point $P$ on the circle. That point $P$ and the two centers form a triangle,
which (due to the problem description) happens to have all three sides
of length $1$. The center of the circle is at the midpoint of the segment
between the centers of the spheres, i.e. at the midpoint of the side
of the triangle opposite $P$. So the radius is the altitude of an
equilateral triangle of side $1$.
