if the points (1,0), (0,0),(0,1),(t,t) are concyclic.then what is the value of t?

Hello Shivam,

A quadrilateral is inscribable in a circle if and only if the product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of opposite sides.

it is obvious from mental diagram that diagonals are (1,0) to (0,1) and (0,0) to (t,t):

d1 = root((1-0)^2 + (0 -1)^2) = root2 
d2 = root((0-t)^2 + (0 -t)^2) = t*root2

and
a1 =  root((1-0)^2 + (0 -0)^2) = 1
a2 =  root((0-t)^2 + (1 – t)^2) =root( t^2 + (t -1)^2)
a3 = root((1-t)^2 + (0 -t)^2) =root( t^2 + (t-1)^2)
a4 =  root((0-t)^2 + (1 -t)^2) = root(t^2 + (t -1)^2)

For concyclic the following conditions are needed to satifsfied:

d1 * d2 = a1*a2 + a3*a4
just keep this pace put values and solve for t and get the answer.
 

Denote the points as O (0,0), A(1,0), B(0,1) and C (t,t). Note that OAB is a right triangle and hence the circle passing through these three points is the one with AB as diameter. which is (x-1/2)²+(y-1/2)² = ½. Putting x=y=t and solving gives t =1 or t=0. Hence C is (1,1)