Question

A circle touches the aside EF of ∆DEF at P and touches sides DE and DF at Q and R resp. when produced. show that DQ=1/2 (perimeter of ∆DEF)​

Answer 1

Given : A circle touches the side's EF of triangle DEF at P and touches Extended sides DE and DF at Q and R respectively.

To Find :  show that DQ=1/2(perimeter of triangle DEF​)

Solution:

perimeter of Δ DEF  =  DE + DF + EF

EF = EP + PF

=> perimeter of Δ DEF  =  DE + DF +  EP + PF

EP = EQ   Equal tangents

FP = FR     Equal tangents

=> perimeter of Δ DEF  =  DE + DF + EQ + FR

=> perimeter of Δ DEF  =  DE+ EQ  + DF + FR

=>  perimeter of Δ DEF  =  DQ  + DR

DQ = DR    Equal tangents

=> perimeter of Δ DEF  =  DQ  + DQ

=>  perimeter of Δ DEF  =  2DQ  

=> (1/2)  perimeter of Δ DEF  =   DQ  

=> DQ = (1/2)  perimeter of Δ DEF

QED

hence proved

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