State "Basic Proportionality Theorem" and use it to prove the follo...

Question

State "Basic Proportionality Theorem" and use it to prove the following :
A line through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

Accepted Answer

Step 1: Basic Proportionality Theorem (BPT / Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then the other two sides are divided in the same ratio. Step 2: Given: In $ riangle ABC$, $D$ is the mid-point of $AB$ and the line $DE \parallel BC$ meets $AC$ at $E$. Step 3: To prove: $E$ is the mid-point of $AC$, i.e. $AE=EC$. Step 4: In $ riangle ABC$, since $DE \parallel BC$, by BPT $$\frac{AD}{DB}=\frac{AE}{EC} \quad \cdots (i)$$ Step 5: Since $D$ is the mid-point of $AB$, $AD=DB$, so $$\frac{AD}{DB}=1 \quad \cdots (ii)$$ Step 6: From (i) and (ii), $$\frac{AE}{EC}=1 \;\Rightarrow\; AE=EC$$ Step 7: Therefore $E$ is the mid-point of $AC$. Hence a line through the mid-point of one side of a triangle, parallel to another side, bisects the third side. Proved.

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