Circle passes through points (1, 2) , (2, 5)

yeah nice approach, but how did u spot the 90 degrees?

I think he spotted it by drawing a diagram without the circle. Or not.

Let's start with (1,2) and (2,5). Draw a right triangle keeping the X-coordinate constant, and we have a third point at (1,5). This gives us three legs of a right triangle. Side A=(1,2) to (1,5) = 3, side B=(1,5) to (2,5) = 1, side C=(2,5) to 1,2)=\(sqrt(A^2+B^2)=sqrt(1^2+3^2)=sqrt(10)\). Use the same method for the points (2,5) and (5,4), but holding the Y-coordinate constant, and the hypotenuse is also \(sqrt(10)\). Applying the same method for points (5,4) and (1,2) you find that the hypotenuse for that triangle is \(sqrt(20)\). Looking at the triangle with the lengths of the legs and hypotenuse calculated you can see that the triangle is a right triangle.

cP

if you see the two sides are sqrt(10),sqrt(10) and third side is sqrt (20)
they are in the ratio 1:1:sqrt(2) .. this possible only for right angle traingle.

this large side must be diameter.

Great job!! Mohindru.

Its nice that you were able to spot the hypoteneuse and the point of right angle. However with this approach, you would be stuck if this is not the case.
The following is an approach which would give result no matter where the points are on the circle. The drawback is that it would take 3-5 min to solve.

Equation of circle is

(x-h)^2 + (y-k)^2 = r^2
where (h,k) are coordinates of center, r = radius and (x,y) are points on circle

Insert the 3 points to obtain the following equations

h^2 + k^2 - 2h -4k + 5 = r^2
h^2 + k^2 - 4h -10k + 29 = r^2
h^2 + k^2 - 10h -8k + 41 = r^2

Solve them and (h,k) = (3,3)
Hence r = sqrt 5
Diameter = 2 sqrt 5 = sqrt 20

On the simira lines, but I did it like this :

Let the coordinate of the center be (x,y ) ; now the distance from all the 3 points will be equal ( radius)

so v[ (x-1)^2 + (y-2)^2] = v[(x-2)^2 + (y-5)^2 ] = v[ (x-5)^2 + y-4)^2 ]

that gives us two equations ( after easy cancellation ) x+3y=12 and 2x+ y= 9 ; which provides us (x,y) = (3,3)

so radius = v [ (3-1)^2 + (3-2)^2 ] = v[ 4] ; so diameter= 2v[4] = v[20]