Three spheres of radii a, b, and c are tangent to the same plane at points A, B, and C respectively. Each sphere is externally tangent to the other two.
If a < b < c, then which internal angle of triangle ABC is largest and what is its value in terms of a, b, and c?
(In reply to
No calculus answer, for the first part by Federico Kereki)
While I agree with Frederico's conclusion, I don't think that one can
argue based on a being very small relative to b and c. In fact, I
think that a cannot be very small relative to b and c.
Using Charlie's formula, the three sides of the triangle are 2sqrt(bc),
2sqrt(ab), and 2sqrt(ac). Because this is a triangle, 2sqrt(bc)
must be less than or equal to 2sqrt(ab) + 2sqrt(ac). Solving for
a shows that a is greater than or equal to (bc)/(b+c+2sqrt(bc)).
If b=c, then a must be greater than or equal to b/4.
re: No calculus answer, for the first part
