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  • In triangle ABC if  are radii of ex-circles and <img alt="r_2=r_1+r_3+r" src="/latex-ima

In triangle ABC if r_1,r_2,r_3 are radii of ex-circles and r_2=r_1+r_3+r then which of following is true? 

Option: 1

A is a acute angle 


Option: 2

B is a right angle


Option: 3

C is a acute angle


Option: 4

All of above


Answers (1)

\\r_2=r_1+r_3+r\\ r_2-r=r_1+r_3\\\\ \frac{\Delta}{(s-b)}-\frac{\Delta}{(s)}=\frac{\Delta}{(s-a)}+\frac{\Delta}{(s-c)}\\\\ \frac{\Delta b}{s(s-b)}=\frac{\Delta (2s-a-c)}{(s-a)(s-c)}\\\\ s(s-b)=(s-a)(s-c)\\ s^2-sb=s^2-s(a+c)+ac\\ s(a+c-b)=ac\\ (a+b+c)(a+c-b)=2ac\\ (a+c)^2-b^2=2ac\\ a^2+c^2=b^2\\

Hence, it is a right angled triangle with right angle at B

Posted by

Sumit Saini

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