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Basic Proportionality Theorem says: If a line is drawn parallel to one side of the triangle to intersect the other two sides at distinct points .Then the other two sides are divided in the same ratio.

PROOF ( to follow this proof, just draw the triangles and segments)

Draw triangle PQR and construct line L parallel to segment QR.

Line L intersects segment PQ and segment PR at S and T respectively.

We want to show that length of PS/ length of QS is equal to length PT/ length of PR since that is what the BPT says.

Construct segments SR and QT.

Look at triangles PTS and QTS and note they have the same height which implies that

the area of triangle PTS/ area of triangle QTS is equal to PS/ SQ.

By the same reasoning, the areas of triangle SPT/ triangle SRT is equal to PT/TR.

Triangles QTS and SRT both have the same height and both have ST as a base segment so they have the same area.

So the ratio of the area of triangle PTS to the area of triangle QTS is equal to the ratios of the area of triangles SPT/SRT.

So the ratio of PS/SQ is equal to PT/TR

Since line L which is parallel to segment QR divides segment PQ and segment PR in the same ratio we have proved the BPT.

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Q: Theory of BPT theorem
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