User:OstadMasum/sandbox

A. R. Khalifa's Formula

Azizur Rahman Khalifa is a former professor of department of mathematics, University of Dhaka. He derived a simpler equation of a circle that can satisfy any three random points on a two dimensional plane.

Let, three random points on a two dimensional plane be P (x$1$,y$1$); Q (x$2$,y$2$) & R (x$3$,y$3$) and a circle to be drawn or the equation of the circle to be determined.

Now, the equation of the circle consisting of the diameter PQ shall be, (x-x$1$)(x-x$2$) + (y-y$1$)(y-y$2$) = 0 ..........(1)

Let, f(x,y) = (x-x$1$)(x-x$2$) + (y-y$1$)(y-y$2$) ∴ f(x,y) = 0 ..........(2) Again, the equation of the straight line PQ be, (x-x$1$)/(x$1$-x$2$) = (y-y$1$)/(y$1$-y$2$) ⇒ (x-x$1$)(y$1$-y$2$) - (x$1$-x$2$)(y-y$1$) = 0 ..........(3) Let, L(x,y) = (x-x$1$)(y$1$-y$2$) - (x$1$-x$2$)(y-y$1$) ∴ L(x,y) = 0 ..........(4) The circle from equation #2 intersects with the straight line from the equation #4 Hence, ∴ f(x,y) - KL(x,y) = 0 ⇒ f(x,y) = KL(x,y) ..........(5) ⇒ K = f(x,y)/L(x,y); which satisfies R (x$3$,y$3$) ∴ from equation #5 we get, K = f(x$3$,y$3$)/L(x$3$,y$3$) ..........(6) When the value of K is replaced, from equation #5 we get, f(x,y) = L(x,y)f(x$3$,y$3$)/L(x$3$,y$3$) ⇒ f(x,y)/f(x$3$,y$3$) = L(x,y)/L(x$3$,y$3$) ⇒ (x-x$1$)(x-x$2$) + (y-y$1$)(y-y$2$)/(x$3$-x$1$)(x$3$-x$2$) + (y$3$-y$1$)(y$3$-y$2$) = (x-x$1$)(y$1$-y$2$) - (x$1$-x$2$)(y-y$1$)/(x$3$-x$1$)(y$1$-y$2$) - (x$1$-x$2$)(y$3$-y$1$) [QED]