Question

In the above figure, the circles with P, Q, and R intersect at points B, C, D, and E as shown. Lines CB and ED intersect in point M. Lines are drawn from point M to touch the circles at points A and F. Prove that MA = MF. 

Answer 1

Given: The circles with P, Q, and R intersect at points B, C, D, and E as shown. Lines CB and ED intersect in point M. Lines are drawn from point M to touch the circles at points A and F.

To Prove: MA = MF

Proof: 

Lines are drawn from point M to touch the circle at point A.

By tangent secant segments theorem,

MA² = MB × MC   ....(i)

By a theorem of the external division of chords,

MB × MC = MD × ME  ...(ii)

Lines are drawn from point M to touch the circle at point F.

MF² = MD × ME  ...(iii)

From (i), (ii), (iii)

MA2 = MF2

Taking square root both the sides,

MA = MF

Hence proved.