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Question

# The points (α,β),(γ,δ)(α,δ) and (γ,β) taken in order, where α,β,γ,δ are different real numbers, are

A
collinear
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B
vertices of a square
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C
vertices of a rhombus
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D
concyclic
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Solution

## The correct option is D concyclicThe given points are not collinear as we can see they form a rectangle.The side's length are not equal which means it is not a square or rhombus.Which only leaves concyclic.Using Ptolemy's theoremProduct of diagonal=Sum of pair of opposite sides then the quadrilateral ca be inscribed in the circleBoth diagonal's d=√(α−γ)2+(β−δ)2Rectangle length(l)=(α−γ) and breadth(b)=(β−δ)d∗d=(α−γ)2+(β−δ)2l∗l=(α−γ)2 and b∗b=(β−δ)2d∗d=l∗l+b∗b Therefore they are concyclic.

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