Prove that the line joining the mid-point of any two sides of a triangle is parallel to the third side.

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Solution

Given: Let us assume $Δ$ ABC where D is the mid point of AB and E is the mid point of AC.

To prove: DE $| |$ BC

Proof:
In $Δ$ ABC,

D is the mid point of AB

$\Rightarrow$ AD $=$ DB

$\Rightarrow \frac{A D}{D B} = 1$ …………….. $(1)$

E is the mid point of AC

$\Rightarrow A E = E C$

$\Rightarrow \frac{A E}{E C} = 1$ …………… $(2)$

From $(1)$ & $(2)$

$\frac{A D}{D B} = \frac{A E}{E C}$

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

$∴ D E | | B C$ .

Hence proved.

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