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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. |