Question

In ΔABC, E and F are mid-points of sides AB and AC respectively. If BF and CE intersect each other at point O,
prove that the ΔOBC and quadrilateral AEOF are equal in area.

Answer 1

E and F are the midpoints of the sides AB and AC.
Consider the following figure.

Therefore, by midpoint theorem, we have, EF || BC

Triangles BEF and CEF lie on the common base EF and between the parallels, EF and BC

Therefore, Ar.( ΔBEF ) = Ar.( ΔCOF )
⇒ Ar.( ΔBOE ) + Ar.( ΔEOF ) = Ar.( ΔEOF ) + Ar.( ΔCOF ) 
⇒ Ar.(ΔBOE ) = Ar.( ΔCOF )

Now BF and CE are the medians of the triangle ABC

Medians of the triangle divide it into two equal areas of triangles.

Thus, we have, Ar. (ΔABF) = Ar. (ΔCBF)

Subtracting Ar. ΔBOE on both the sides, we have

Ar. (ΔABF) - Ar. (ΔBOE) = Ar. (ΔCBF) - Ar. (ΔBOE)

Since, Ar. ( ΔBOE ) = Ar. ( ΔCOF ),

Ar. (ΔABF) - Ar. (ΔBOE) = Ar. (ΔCBF) - Ar. (ΔCOF)

Ar. ( quad. AEOF ) = Ar. ( ΔOBC ) , hence proved