Question

'O' is a point in the triangle ABC OA, OB and OC are jointly and
produced to meet BC,CA and
AB at D, E and F respectively, then
then value of OD/AD+OE/BE+OF/CF​
(1)4 (2)3
(3)2 (4)1

Answer 1

Given :-

• O is a point in the triangle ABC.
• OA, OB and OC are jointly and produced to meet BC,CA and AB at D, E and F respectively.

To Find :-

• OD/AD + OE/BE + OF/CF = ?

Solution :-

First lets understand what is a cevian :-

• cevian is a line segment in a triangle, joining any vertex of the triangle with any one point on its opposite side .
• we can say that all medians, altitudes, angle bisectors of any triangle can be referred as Cevians.
• from image we have , AD, BE and CF are cevians of ∆ABC.

There are some Properties of the Ratios of Lengths formed by Three cevians , when all passing through the same interior point. (in question we have given that, O is the point of intersection of all three cevians.) .

These Properties are :-

1. (AF/FB) * (BD/DC) * (CE/EA) = 1 .
2. AO/OD = AE/EC + AF/FB.
3. OD/AD + OE/BE + OF/CF = 1 .
4. AO/AD + BO/BE + CO/CF = 2 .

Therefore,

→ OD/AD + OE/BE + OF/CF = 1 . (Option 4.) (Ans.)

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