Show-that-if-z-1-z-2-z-3-z-4-0-and-z-1-z-2-0-then-the-complex-numbers-z-1-z-2-z-3-z-4-are-concyclic-

Question Number 20549 by Tinkutara last updated on 28/Aug/17

S h o w t h a t i f z_{1} z_{2} + z_{3} z_{4} = 0 a n d z_{1} +

z_{2} = 0, t h e n t h e c o m p l e x n u m b e r s z_{1},

z_{2}, z_{3}, z_{4} a r e c o n c y c l i c .

Commented by ajfour last updated on 28/Aug/17

Answered by ajfour last updated on 28/Aug/17

O r i g i n i s t h e m i d p o i n t o f t h e j o i n

o f z_{1} a n d z_{2} . (a s z_{1} + z_{2} = 0)

z_{3} z_{4} + z_{1} z_{2} = 0 \Rightarrow z_{3} z_{4} = z_{1}^{2}

\Rightarrow (\frac{z_{4}}{z_{1}}) (\frac{z_{3}}{z_{1}}) = 1

\Rightarrow a r g (\frac{z_{4}}{z_{1}}) = - a r g (\frac{z_{3}}{z_{1}})

a l s o ∣ z_{3} ∣∣ z_{4} ∣=∣ z_{1} ∣^{2}

o r ∣ z_{5} ∣∣ z_{4} ∣=∣ z_{1} ∣^{2}

\Rightarrow x y = R^{2}

p o w e r o f p o i n t (h e r e o r i g i n) i s

t h e s a m e f o r c h o r d j o i n i n g z_{1}, z_{2}

a n d c h o r d j o i n i n g z_{4}, z_{5} .

a n d a s ∣ z_{3} ∣=∣ z_{5} ∣; z_{1}, z_{2}, z_{3}, z_{4} a r e

c o n c y c l i c .

Commented by Tinkutara last updated on 28/Aug/17

Thank you very much Sir!

Commented by Tinkutara last updated on 28/Aug/17

Why ∣ z_{3} ∣ = ∣ z_{5} ∣ ? How you have defined

z_{5} ?

Commented by ajfour last updated on 28/Aug/17

i t i s r e f l e c t i o n o f z_{3} a b o u t t h e

p e r p e n d i c u l a r b i s e c t o r o f t h e j o i n

o f z_{1} a n d z_{2} . H e n c e ∣ z_{5} ∣=∣ z_{3} ∣ .

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