# 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.

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