Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.
Let ∆ABC be any triangle such that O is the origin.
∴Let coordinates be A(0, 0), B(x1 , y1), C(x2 , y2).
Let D and E are the mid-points of the sides AB and AC respectively.
We have to prove that line joining the mid-point of any two sides of a triangle is equal to half of the third side which means,
DE = 
BC
By midpoint formula,
x = 
, y = 
For midpoint D on AB,
x =
, y = 
∴ x = 
and y = 
∴ Coordinate of D is (
, 
)
For midpoint E on AC,
x =
, y = 
∴ x = 
and y = 
∴ Coordinate of E is ( 
, 
)
By distance formula,
XY = 
For BC,
BC = 
For DE,
DE = 
=
( 
)
= 
BC
∴ DE = 
BC
Hence, we proved that line joining the mid-point of any two sides of a triangle is equal to half of the third side.