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

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Solution

Consider the following figure In this $P Q$ is a line segment joining midpoint of $A B$ and $A C$
i.e
$A P = P B a n d A Q = Q C$

$\frac{A P}{P B} = 1$ and $\frac{A Q}{Q C} = 1$

$\frac{A P}{P B}$ = $\frac{A Q}{Q C}$

Hence using proportionality theorem we obtain.

if $\frac{A P}{P B} = \frac{A Q}{Q C}$

Therefore $P Q | | B C$

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

Consider the following figure In this PQ is a line segment joining midpoint of AB and ACi.eAP=PB and AQ=QCAPPB=1 and AQQC=1APPB =AQQCHence using proportionality theorem we obtain.if APPB=AQQCTherefore PQ||BC