Question

Prove that the equation of a circle is
(x − a)(x − b) + (y − c)(y − d) = 0
if the endpoints of its diameter are located at (a,c) and (b,d).

nonsense = report

Answer 1

Answer:

You have an easier job using Pythagoras on the sides of the triangle:

[(x−a)2+(y−b)2]+[(x−c)2+(y−d)2]=(a−c)2+(b−d)2

or (noting that the squares of constants cancel and dividing though by 2)

x2−ax−cx+y2−bx−dx=−ac−bd

from which the conclusion follows.

[But the other way using the scalar/dot product is neater and shows more insight]