Inscribed Quadrilateral ?

1 Answer
Aug 26, 2016

The quadrilateral is a regular quadrilateral.

Explanation:

If the points lie on a circle with radius 1 then they can be represented as

eiϕ1+eiϕ2+eiϕ3+eiϕ4=0

or equivalently

sinϕ1+sinϕ2+sinϕ3+sinϕ4=0

and

cosϕ1+cosϕ2+cosϕ3+cosϕ4=0

(Here we used de Moivre's identity eiϕ=cosϕ+isinϕ )

grouping and squaring both sides

(sinϕ1+sinϕ2)2=((sinϕ3+sinϕ4))2
(cosϕ1+cosϕ2)2=((cosϕ3+cosϕ4))2

and adding side by side we get at

2+2(sinϕ1sinϕ2+cosϕ1cosϕ2)=2+2(sinϕ3sinϕ4+cosϕ3cosϕ4)

or

cos(ϕ2ϕ1)=cos(ϕ4ϕ3)

grouping now

(sinϕ2+sinϕ3)2=((sinϕ1+sinϕ4))2
(cosϕ2+cosϕ3)2=((cosϕ1+cosϕ4))2

we get at

cos(ϕ3ϕ2)=cos(ϕ4ϕ1)

So we can conclude that the quadrilateral is a regular quadrilateral.