Problem of a circle tangent to three other circles

Question

Two circles with centres A and B and radii 14 and 7 units respectively touch each other externally. M is the mid point of segment DE and is the centre of the circle with radius 21 units. The two smaller circles touch the larger circle internally. A co ordinate system has been set up with the origin as M, and other points lie on the X-axis.

To find: The coordinates of the centre and radius of a circle and which touches the smaller circles externally and the larger circle internally.

Note: To be solved without using Apollonius problem and Descartes theorem.

Hints: Use Stewart's theorem and Pythagoras theorem

Answer 1

Assume the radius of the circle which touches the smaller circles externally and the larger circle internally as r and name it as C. Join AC,BC and CM and extend it to circumference at N . so, AC= r+14, BC=7+r, AM=7, MB=14, MN(radius)=21, MN=MC+r,so,MC=MN-r,MC=21-r.

In triangle, ABC, ar(ACM)/ar(MCB)=AM/MB -{1}
{AM/MB=1:2}

Now find the area of ACM and area of MCB from heron's formula in terms of r.

put it in eq.{1} and get your answer.....