Question

in the adjoining figure the circles with centre P,Q,R intersect at points B, C, D and E as shown. Line CB and line ED intersect at point M. Lines drawn from point M touch the circle at points A and F. Prove that MA=MF

Answer 1

Explanation:

Given that there are 3 circles with centres P,Q,R intersect at the points B, C, D and E

Line CB and line ED intersect at point M. Also, given that lines drawn from point M touch the circle at points A and F.

We need to prove that MA = MF

Let us consider the circle with centre P

From the figure,it is obvious that MA is tangent and MC is secant.

Then, by tangent secant theorem, we have,

$MA^2=MB\times MC$ -------------(1)

Now, we shall consider the circle with centre Q.

From the figure, we can see that the lines MC and ME intersect externally at the point M.

Then, by external division of chord theorem, we have,

$MC\times MB=ME\times MD$ --------(2)

Considering the circle with center R, MF is the tangent and ME is the secant.

Then, by tangent secant theorem, we have,

$MF^2=MD\times ME$ -----------(3)

Substituting equation (1) and (3) in (2), we get,

$MA^2=MF^2$

Taking square root on both sides, we have,

$MA=MF$

Hence proved

Learn more:

(1) In the given figure common tangents A B and C D to the two circle intersect at E. prove that a b is equal to CD

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(2) Tangent secant theorem​

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