Question

The bisector of the right ZA of the right angled triangle AABC cuts BC
at D and the perpendicular from D cuts AC at E. Prove that DE ® (AB
+ AC) = AB X AC.​

Answer 1

Answer:

ANSWER

Given, internal angle bisector of A meets BC at D,

we know length of AD is given by, AD = b+c2bccos2A

Given, △ADE is right angled triangle with right angle at D. 

∠D=90∘,∠AED=90−2A

Using sine rule in △ADE, we get

sin∠AEDAD=sin∠ADEAE=sin∠2ADE

cos2AAD=AE=sin2ADE

Substitute the value of AD in above equation, we get

AE=cos2Ab+c2bccos2A=b+c2bc

so, AE is harmonic mean of b and c. 

Consider △ADE and △ADF , we know that

∠ADE=∠ADF=90∘∠DAE=∠DAF=∠2

Answer 2

Answer:

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Step-by-step explanation:

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