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Three spheres are tangent to a plane at the vertices of a triangle and are tangent to each other. find the radii of these spheres if the sides of the triangle are 6, 8, and 10.

User Holstebroe
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Three spheres are tangent to a plane at the vertices of a triangle and are tangent to each other. Denote the points where spheres are tangent to the lane A, B, C. You get a triangle ABC with lengths of sides 6, 8 and 10. Denote
C_A, C_B, C_C the projections of the tangent points between spheres on the pane.
Then

CC_B=CC_A=c', \\ AC_B=AC_C=a', \\ BC_A=BC_C=b'
and

a'+b'=6, \\ a'+c'=8, \\ b'+c'=10.

Add all these three equations:
2a'+2b'+2c'=24, a'+b'+c'=12 and substract each equation from
a'+b'+c'=12:


a'+b'+c'-(a'+b')=12-6, \\ a'+b'+c'-(a'+c')=12-8, \\ a'+b'+c'-(b'+c')=12-10.
Then

c'=6, \\ b'=4, \\ a'=2 are spheres' radii.




Three spheres are tangent to a plane at the vertices of a triangle and are tangent-example-1
User Rozwel
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