In three line segments OA, OB, and OC, points L, M, N respectively are so chosen that and but neither of L, M, N nor of A, B, C are collinear. Show that .

We have LM∥AB and MN∥BC

by the converse of proportionality theorem

OL/AL=OM/MB ……….(i)

ON/NC=OM/MB ………(ii)

Comparing equ.(i)and(ii)

OL/AL=ON/NC

Thus LN divides side OA and OC of ⊿ OAC in same ratio

Then by the converse of basic proportionality theorem