Question

If three medians of a triangle are equal prove that it is an isosceles triangle

Answer 1

To prove: The medians of an equilateral triangle are equal.

Median = The line joining the vertex and mid-points of opposite sides.

Proof: Let Δ ABC be an equilateral triangle

AD, EF and CF are its medians.

Let,

AB = AC = BC = x

In

BFC and

CEB, we have

AB = AC (Sides of equilateral triangle)

AB =

AC

BF = CE

∠ABC =∠ACB (Angles of equilateral triangle)

BC = BC (Common)

Hence, by SAS theorem, we have

Δ BFC ≅ Δ CEB

BE = CF (By c.p.c.t)

Similarly, AB = BE

Therefore, AD = BE = CF

Hence, proved