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Transcript
Theorem 12 Let ABC be a triangle . If the line t is parallel to BC and cuts [AB] in the ratio m:n, then it also cuts [AC] in the same ratio. USE THE FORWARD AND THE BACK ARROWS ON THE KEYBOARD TO VIEW AND REWIND PROOF. Given: Triangle ABC Line t is parallel to BC Line t cuts AB in the ratio m:n at D To Prove: Line t also cuts AC in the ratio m:n. Construction: Divide AB into m+n equal segments. (The number of divisions depends on the value of m and n.) Proof: Line t cuts AB in the ratio m:n at D. (Where m and n are natural numbers.) Thus there are equally spaced points D0=B, D1, D2, D3 , ...... Dm-1,D m= D, Dm+1,.....Dm+n-1, Dm+n=A (The number of divisions depends on the value of m and n.) Note the segments [B, D1], [D1,D2] etc. are equal in length. Draw lines D1 E 1, D2 E 2 ,,..... parallel to BC with E 1, E 2 etc on AC. Then [CE1 ], [E1E 2 ], [E 2 E3 ],........ [Em+n-1 A ] have the same length. Theorem 11 Hence E divides AC in the ratio m:n and line t cuts AC in the ratio m:n. Q.E.D. © Project Maths Development Team