What makes four points concyclic?

Geometry Level 2

If ( m i , 1 m i ) \left(m_i,\dfrac{1}{m_i}\right) where i = 1 , 2 , 3 , 4 i=1,2,3,4 are the coordinates of four concyclic points, then what is the value of m 1 m 2 m 3 m 4 m_1m_2m_3m_4 ?

∞ \infty 1 1 − 1 -1 0 0

Nihar Mahajan by Nihar Mahajan , 20, India

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1 solution

Saarthak Marathe
May 26, 2016

Let the general equation of a circle be,

x 2 + y 2 + 2 g x + 2 f y + c = 0 {x}^2+y^2+2gx+2fy+c=0

Now put x = m , y = 1 / m x=m,y=1/m

Then the equation becomes

m 4 + 2 g m 3 + c m 2 + 2 f m + 1 = 0 m^4+2gm^3+cm^2+2fm+1=0

As m 1 , m 2 , m 3 , m 4 {m}_{1},{m}_{2},{m}_{3},{m}_{4} are roots of this equation,

By Vieta's formula

m 1 m 2 m 3 m 4 = 1 {m}_{1}{m}_{2}{m}_{3}{m}_{4}=1

21 Helpful 0 Interesting 0 Brilliant 0 Confused

Same way. Unfortunately, I can't type fast.

A Former Brilliant Member - 5 years ago

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