Area of Δ ABC = 1/2 Base * altitude
= 1/2 AC * BE = 1/2 AB * CF = 1/2 AB * BE, given BE = CF
So AC = AB
Hence it is an isosceles Δ.
Consider the triangles ABE and ACF.
Angle A is common. angle BEA = 90 = angle CFA. The triangles are similar.
So isosceles triangle.