Remember me: tan A+tan B+tan C=tan A tan B tan C ?

Geometry Level pending

If x,y and z be the lengths of the perpendiculars dropped from the circumcenter to the sides BC,CA and BA respectively of a triangle ABC,the reciprocal value of the expression x y z / a b c ( a / x + b / y + c / z ) xyz/abc(a/x+b/y+c/z) is ?

16 4 2 8

Sanchayan Dutta by Sanchayan Dutta , 23, India

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1 solution

Sanchayan Dutta
Sep 15, 2015

Let O be the circumcenter.R,P,Q be the foot of perpendiculars from O on AB,BC,CA respectively.Let OP=x,OQ=y,OR=z.

t a n A = a / 2 x tan A=a/2x (since angle BOP is equal to A)

Similarly t a n B = b / 2 y tan B=b/2y and t a n C = c / 2 z tan C=c/2z

Now for a triangle tan A+tan B+tan C=tan A tan B tan C

Hence,

a / 2 x + b / 2 y + c / 2 z = a b c / 8 x y z a/2x+b/2y+c/2z=abc/8xyz a / x + b / y + c / z = a b c / 4 x y z a/x+b/y+c/z=abc/4xyz

We see the required answer is 4.

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