Spheres and planes Watch

dt003
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Three spheres of radii 1,2, 3 units, touch each other externally. Prove that a plane which touches each sphere makes an angle \arcsin (\frac{1}{6} \sqrt{13}) with the plane containing the centers of the spheres.

I am not sure how to approach this. I was thinking of fixing the sphere of radius 1 to the origin of a coordinate system and finding position vectors of the other two centers so that I could find the vector normal to the plane through the centers. I noticed that the triangle through the centers of the spheres is right, I don't know if this will help. For the plane tangent to all 3 spheres I am stuck. I worked out that any point on a sphere radius c center with position vector \vec{a} satisfies (\vec{r}-\vec{a}) \cdot (\vec{r}-\vec{a}) = c^2 but I don't know how to make it work in this problem where each sphere touches the other two. All I need is the two vectors normal to the two planes.

I was wondering if the intersections of the spheres are coplanar with the centers.
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ghostwalker
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(Original post by dt003)
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I'd put the centres of the spheres in the x,y plane, and as you've worked out the centres make a 3,4,5 triangle, you can work out positions of the three centres (stick them on the axes).

Now consider the plane contacting the outside of the three spheres. What's its form (general coordinate geometry form for a plane)? And what's the distance of a point from a plane (for each of the three centres)?

Not worked it through, but that should lead you in the right direction(!).
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dt003
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(Original post by ghostwalker)
I'd put the centres of the spheres in the x,y plane, and as you've worked out the centres make a 3,4,5 triangle, you can work out positions of the three centres (stick them on the axes).

Now consider the plane contacting the outside of the three spheres. What's its form (general coordinate geometry form for a plane)? And what's the distance of a point from a plane (for each of the three centres)?

Not worked it through, but that should lead you in the right direction(!).
Yes! I got it.
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