Is it true that the area of a triangle is 4 times the area of the triangle formed by joining the midpoints of the three sides? How? What is the proof?

Hello student,
Please find the answer to your question below
Let DEF be the midpoints of sides of a triangle ABC( with D on BC, E on AB and F on AC ). Now, considering triangles AEF and ABC, angles EAF = BAC and AE / AB = 1/2 and AF/AC = 1/2.
Hence, both triangles are similar by the SAS ( Side - Angle - Side ) criterion and correspondingly as AE/AB=AF/AC=EF/BC ( similar triangle properties ), EF =BC/2.
the cases DF=AC/2 and DE=AB/2 can be proved in the same way.
Now, triangle DEF and ABC are similar with DE/AB=EF/BC=DF/AC =1/2.
Similar triangles have the property :
Ratio of areas = square of ratios of sides
and hence, ( area of triangle DEF )/( area of triangle ABC ) = 1/2*1/2 = 1/4.

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