As shown in the diagram below, circle D is internally tangent to circle A...............what is the radius of circle D?

Question

As shown in the diagram below, circle D is internally tangent to circle A and externally tangent to circles B and C.

If circle B has a radius of 2 and C has a radius of 1, what is the radius of circle D?

Answer 1

6/7

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See pic. It is self-explanatory.
Blue, red and green lines have values as shown above.
BA=1, AC=2
On the triangle BDC, name angle C by a.
By applying Al-Kashi’s law of cosines to triangles BDC and ADC we get the following two equations:
(2 + r)2 = (1 + r)2 + 32 – 2(1 + r)(3)cos a,
(3 – r)2 = (1 + r)2 + 22 – 2(1 + r)(2)cos a,
Solving which we get: r = 6/7