# A circle touches

Given that a circle touches the sides BC, CA and AB of ΔABC at points D, E and F respectively.

Let BD = x, CE = y and AF = z.

We have to prove that area of ΔABC =

Proof:

BD and BF are tangents drawn from B. And D and F are points of contact.

BD = BF = x

Similarly for tangents CE and CD drawn by C and tangents AE and AF drawn from A,

CE = CD = y and AF = AE = z

Sides of ΔABC,

AB = c = AF + BF = z + x … (1)

BC = a = BD + DC = x + y … (2)

CA = b = CE + AE = y + z … (3)

In ΔABC, 2s = AB + BC + AC

= (z + x) + (x + y) + (y + z)

= 2 (x + y + z)

s = x + y + z … (4)

We know that area of ΔABC =

Area =

=

=

Hence proved.

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