Question

In the adjoining figure the circles with centres p,q and r intersect at points b,c,d and e as shown.lines cb and ed intersect at point m​

Answer 1

In the adjoining figure the circles with centres p,q and r intersect at points b,c,d and e as shown.lines cb and ed intersect at point m​

From figure it's clear that,

MA is a tangent

MC is a secant with center P.

Using the tangent - secant theorem, we get,

MA² = MB × MC  ..............A

Similarly, we have,

MF is a tangent

ME is a secant with center R.

Using the tangent - secant theorem, we get,

MF² = MD × ME .............B

In Δ MCD and Δ MEB,

∠ MCD = ∠ MEB  (angles inscribed from same arc)

∠ CMD = ∠ EMB (common angle)

∴ Δ MCD ~ Δ MEB ( using AA theorem criteria)

⇒ MC/ME = MD/MB  (using C.S.S.T)

∴ MC × MB = MD × ME   ...........C

Using A, B and C, we get,

MA² = MF²

∴ MA = MF

Similarly,

MC = ME

Therefore, it is proved that, CB and ED intersect at point M.