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O is any point inside a triangle ABC. The bisector of ∠AOB, ∠BOC and ∠COA meet the sides AB, BC and CA in point D, E and F respectively. Show that AD x BE x CF = DB x EC x FA.

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In ΔAOB, OD is the bisector of ∠AOB.

∴ OA/OB = AD/DB ... (1)

 In ΔBOC, OE is the bisector of ∠BOC

∴ OB/OC = BE/EC … (2)

In ΔCOA, OF is the bisector of ∠COA.

∴ OC/OA = CF/FA … (3)

Multiplying the corresponding sides of (1), (2) and (3), we get

⇒ DB x EC x FA = AD x BE x CF

Hence proved.

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