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| T-N-T (Posted on 2017-11-25) |
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Let ABC be an arbitrary triangle with points D, E, and F
lying on rays AB, BC, and CA respectively such that
AD/AB = BE/BC = CF/CA = x.
Prove that Area(ΔDEF)/Area(ΔABC) = 3x2 - 3x + 1.
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Submitted by Bractals
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Solution:
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Area(ΔDEF) = ½| FD ⊗ FE | =
½| [xAB + (x - 1)AC] ⊗ [(1 - x)AB + (2x - 1)AC] | =
½ [x(2x - 1) - (x - 1)(1 - x)] | AB ⊗ AC | =
[ 3x2 - 3x + 1 ] * Area(ΔABC)
QED
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