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Question
The triangle whose vertices are the points represented by the complex numbers z1,z2,z3 on the Argand diagram is equilateral. Then show that 1z2−z3+1z3−z1+1z1−z2=0
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Solution
Let ABC be the given triangle, where the points A, B and C represent complex numbers z1,z2 & z3 respectively. If the triangle ABC is equilateral, then AB=BC=CA.
∴|z1−z2|=|z2−z3|=|z3−z1|=k ....... (i)
Now, 1z1−z2=¯¯¯¯¯z1−¯¯¯¯¯z2(z1−z2)(¯¯¯¯¯z1−¯¯¯¯¯z2)=¯¯¯¯¯z1−¯¯¯¯¯z2|z1−z2|2=¯¯¯¯¯z1−¯¯¯¯¯z2k2 {from (i)}
Similarly 1z2−z3=¯¯¯¯¯z2−¯¯¯¯¯z3k2
and 1z3−z1=¯¯¯¯¯z3−¯¯¯¯¯z1k2∴1z2−z3+1z3−z1+1z1−z2=0
Conversely :
1z2−z3+1z3−z1+1z1−z2=0
On multiplying by (z3−z1)
⇒z3−z1z2−z3+1+z3−z1z1−z2=0
⇒z2−z1z2−z3+z3−z1z1−z2=0
⇒arg(z2−z1z2−z3)=arg(z3−z1z2−z1)⇒∠B=∠A
Similarly ∠A=∠C⇒∠A=∠B=∠C=π/3. Hence the triangle is equilateral.
Ans: 1
∴|z1−z2|=|z2−z3|=|z3−z1|=k ....... (i)
Now, 1z1−z2=¯¯¯¯¯z1−¯¯¯¯¯z2(z1−z2)(¯¯¯¯¯z1−¯¯¯¯¯z2)=¯¯¯¯¯z1−¯¯¯¯¯z2|z1−z2|2=¯¯¯¯¯z1−¯¯¯¯¯z2k2 {from (i)}
Similarly 1z2−z3=¯¯¯¯¯z2−¯¯¯¯¯z3k2
and 1z3−z1=¯¯¯¯¯z3−¯¯¯¯¯z1k2∴1z2−z3+1z3−z1+1z1−z2=0
Conversely :
1z2−z3+1z3−z1+1z1−z2=0
On multiplying by (z3−z1)
⇒z3−z1z2−z3+1+z3−z1z1−z2=0
⇒z2−z1z2−z3+z3−z1z1−z2=0
⇒arg(z2−z1z2−z3)=arg(z3−z1z2−z1)⇒∠B=∠A
Similarly ∠A=∠C⇒∠A=∠B=∠C=π/3. Hence the triangle is equilateral.
Ans: 1
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