Question

The points $(\alpha ,\beta ),(\gamma ,\delta )(\alpha ,\delta )$ and $(\gamma ,\beta )$ taken in order, where $\alpha ,\beta ,\gamma ,\delta$ are different real numbers, are

• A. collinear
• B. vertices of a square
• C. vertices of a rhombus
• D. concyclic

Answer 1

The correct option is D concyclic

The 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 theorem

Product of diagonal=Sum of pair of opposite sides then the quadrilateral ca be inscribed in the circle

Both diagonal's $d=\sqrt{(\alpha -\gamma {)}^2+(\beta -\delta {)}^2}$

Rectangle $length\left(l\right)=(\alpha -\gamma )$ and $breadth\left(b\right)=(\beta -\delta )$

$d\ast d=(\alpha -\gamma {)}^2+(\beta -\delta {)}^2$

$l\ast l=(\alpha -\gamma {)}^2$ and $b\ast b=(\beta -\delta {)}^2$

$d\ast d=l\ast l+b\ast b$ Therefore they are concyclic.