AP EAPCET 05-Jul-2022 Shift 2 Paper

Let 0 , the circumcentre of circle be taken as origin.Let a,b and c are the position vector of the vertices A,B and C, respectively.
Position vector of G=(

a+b+c

3

)
Let BC=a,AB=c and AC=b
‌|BC|=a
⇒|c−b|=a
⇒(c−b)2=a2
⇒|c|2+|b|2−2b⋅c=a2
⇒R2+R2−2b⋅c=a2
⇒2b⋅c=2R2−a2
Similarly, 2a⋅b=2R2−c2 and 2a⋅c=2R2−b2
‌OG=‌

a+b+c

3

‌|OG|2=‌

(a+b+c)2

9

=‌‌

9R2−1(a2+b2+c2)

9

=‌R2−‌

1

9

(a2+b2+c2)