Question

E and F are the midpoints of side AB, AC of a triangle CE, BF are produced to X and Y. Prove that XAY is a straight line

Answer 1

Answer:

E and F are the midpoints of the sides AB and AC.

Consider the following figure.

Therefore, by midpoint therom, we have EF∥BC

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

Therefore, Ar.(△BEF)=Ar.(△CEF)

⇒Ar.(△BOE)+Ar.(△EOF)=Ar.(△EOF)+Ar.(△COF)

⇒Ar.(△BOE)=Ar.(△COF)

Now BF and CE aare the medians of the triangle ABC

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

Thus, we have Ar.△ABF=Ar.△CBF

Subtracting Ar.△BOE on the both the sides, we have

Ar.△ABF−Ar.△BOE=Ar.△CBF−Ar.△BOE

Since, Ar.(△BOE)=Ar.(△COF)

Ar.△ABF−Ar.△BOE=Ar.△CBF−Ar.△COF

Ar.(quad.AEOF)=Ar.(△OBC), hence proved