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# Basic Proportionality Theorem Proof

If a given line passes through the two sides of the given triangle and parallel to the third side, then it cuts the sides proportionally. This is called the Basic Proportionality theorem..

## Basic Proportionality Theorem Proof

STATEMENT
A line parallel to one side of a triangle divides the other two sides into parts of equal proportion..

Given:

In triangle ABC, a line drawn parallel to BC cuts AB and AC at P and Q respectively.

Image/Diagram:

To Prove:

AP/PB = AQ/QC

Let the point P divide AB in the ratio of l: m where l and m are natural numbers. Divide AP into 'l' and PB into 'm' equal parts. Through each of these points on AB, draw lines parallel to BC to cut AC.

PROOF:

 Step 1: Cut AP into equal parts and draw lines through these points parallel to BC Step 2: By intercept theorem,AQ is cut into equal parts Step 3: By Step 1,QC is cut into m equal parts Step 4: By construction,AP/PB=l/m Step 5: By step 2 and 3,AQ/QC=l/m Step 6: By step 4 and 5,AP/PB=AQ/QC

Hence the proof..

If a given line passes through the two sides of the given triangle and parallel to the third side, then it cuts the sides proportionally. This is called the Basic Proportionality theorem..