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proof of THALES THEOREM

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.



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.



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