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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

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closed as off-topic by heropup, Claude Leibovici, JonMark Perry, user91500, user99914 Feb 3 '16 at 10:48

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  • $\begingroup$ Where does DE come into play? $\endgroup$ – Sawarnik May 15 '14 at 5:55
  • $\begingroup$ And can you include a figure? $\endgroup$ – Sawarnik May 15 '14 at 6:06
  • $\begingroup$ I've added a diagram according to my understanding of the problem; @Chinmay, if this is incorrect, please let us know. $\endgroup$ – user21467 May 16 '14 at 22:50
  • $\begingroup$ If the figure, is in fact correct, the following video might have some ideas for an alternate solution: youtube.com/watch?v=sG_6nlMZ8f4 $\endgroup$ – Alice Ryhl May 16 '14 at 23:01
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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.....

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