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Question
If A , B , C , D , four points in a plane , prove that AD . BC ≤ BD . CA + CD . AB
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Solution
Let z_1 , z_2 , z_3 , z_4 be the affixes of the points A , B , C , D , we have the identity
(z1−z4)(z2−z3)+(z2−z4)(z3−z1)+(z3−z4)(z1−z2)=0
or|−(z1−z4)(z2−z3)|=|(z2−z4)(z3−z1)+(z3−z4)(z1−z2)|
or|(z1−z4)(z2−z3)|≤|(z2−z4)(z3−z1)+(z3−z4)(z1−z2)|
or|(z1−z4)||(z2−z3)|≤|(z2−z4)||(z3−z1)|+|(z3−z4)||(z1−z2)|
i.e. AD⋅BC≤BD⋅CA+CD⋅AB
(z1−z4)(z2−z3)+(z2−z4)(z3−z1)+(z3−z4)(z1−z2)=0
or|−(z1−z4)(z2−z3)|=|(z2−z4)(z3−z1)+(z3−z4)(z1−z2)|
or|(z1−z4)(z2−z3)|≤|(z2−z4)(z3−z1)+(z3−z4)(z1−z2)|
or|(z1−z4)||(z2−z3)|≤|(z2−z4)||(z3−z1)|+|(z3−z4)||(z1−z2)|
i.e. AD⋅BC≤BD⋅CA+CD⋅AB
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