SOLUTION: Inside a circle, with centre O and radius r, two circles with centres A and B are drawn, which touch each other externally and the given circle internally. Prove that the perimeter

Algebra ->  Geometry-proofs -> SOLUTION: Inside a circle, with centre O and radius r, two circles with centres A and B are drawn, which touch each other externally and the given circle internally. Prove that the perimeter      Log On


   

Question 1137177: Inside a circle, with centre O and radius r, two circles with centres A and B are drawn, which touch each other externally and the given circle internally. Prove that the perimeter of the triangle AOB is 2r.
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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1.  Make a sketch.


2.  Let C be the point where the two small circles touch each other.


    Let  |AC| = x  (the unknown length - same as the radius of each of the two small circles).

    Then |BC| = x,  too.

    |AO| = r-x  and  |BO| = r-x    (OBVIOUS.  Use the sketch)


    The perimeter of the triangle AOB is equal to

        |AO| + |BO| + |AC| + | BC| = (r-x) + (r-x) + x + x = 2r.

QED.
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Proved, explained, solved and completed.