Question

In the figure in the adjoining figure the circles with Centre p q and R intersect at point B C D and E line CB and E
D intersect at point M lines drawn from point M touch the circle at point A and F prove that MA is equal to MF

Answer 1

Explanation:

Given that P,Q, R intersect at the points B,C,D and E

The lines CB and ED intersect at the point M.

From point M lines drawn touch the circle at point A and F

To prove : MA = MF

Consider the circle with center P

The line MA is a tangent and the line MC is secant.

By tangent secant theorem, we have,

$M A^{2}=M B \times M C$ -------(1)

Now, let us consider the circle with center Q.

The lines MC and ME that intersect externally at the point M.

By external division of chord theorem, we get,

$M C \times M B=M E \times M D$ ---------(2)

Consider the circle with center R.

The line MF is a tangent and ME is secant.

By tangent secant theorem, we have,

$M F^{2}=M D \times M E$ --------(3)

Using the equations (1) and (3) in the equation (2), we get,

$M A^{2}=M F^{2}$

Taking square root, we get,

$M A=M F$

Thus, the theorem is proved.

Learn more:

(1) 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

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(2) 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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