If ‘O’ is circumcentre and H is orthocenter, G is the centroid of $\mathrm{\Delta ABC}$ , then

	List I		List II
(A)	$\overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}$	(P)	$\frac{1}{2}\overline{\mathrm{HO}}$
(B)	$\overline{\mathrm{HA}}+\overline{\mathrm{HB}}+\overline{\mathrm{HC}}$	(Q)	$2\overline{\mathrm{HO}}$
(C)	$\overline{\mathrm{AH}}+\overline{\mathrm{HB}}+\overline{\mathrm{HC}}$	(R)	$2\overline{\mathrm{AO}}$
(D)	$\overline{\mathrm{OG}}$	(S)	$\frac{1}{3}\overline{\mathrm{OH}}$
		(T)	$\overline{\mathrm{OH}}$

	(A)	(B)	(C)	(D)
(1)	T	Q	S	P
(2)	T	Q	R	S
(3)	T	Q	S	R
(4)	T	Q	R	P

D is mid point of BC
$\mathrm{\Delta AHG},\mathrm{\Delta DOG}$are similar  $\frac{\mathrm{AH}}{\mathrm{OD}}=\frac{\mathrm{HG}}{\mathrm{GO}}=\frac{2}{1}\Rightarrow \overline{\mathrm{AH}}=2\overline{\mathrm{OD}}$
(I)  $\overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}=\overline{\mathrm{OA}}+\overline{\mathrm{AH}}=\overline{\mathrm{OH}}$
(II) $\overline{\mathrm{HA}}+\overline{\mathrm{HB}}+\overline{\mathrm{HC}}=\overline{\mathrm{OA}}+\overline{\mathrm{AB}}+\overline{\mathrm{OC}}-3\overline{\mathrm{OH}}=\overline{\mathrm{OH}}-3.\overline{\mathrm{OH}}=-2\overline{\mathrm{OH}}=2\overline{\mathrm{HO}}$
(III)$\overline{\mathrm{AH}}+\overline{\mathrm{HB}}+\overline{\mathrm{HC}}=\overline{\mathrm{AH}}-\overline{\mathrm{HA}}+\overline{\mathrm{HA}}+\overline{\mathrm{HB}}+\overline{\mathrm{HB}}+\overline{\mathrm{HC}}=2\overline{\mathrm{AH}}+2\overline{\mathrm{HO}}=2\overline{\mathrm{AO}}$
(IV) $\overline{\mathrm{OG}}=\frac{\overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}}{3}=\frac{1}{3}\overline{\mathrm{OH}}$