## Equation of a circle

Consider a fixed complex number and let be any complex number which moves in such a way that its distance from is always equal to r. This implies would lie on a circle whose centre is and radius r. And, its equation would be

or

or ,

or

Let and . Then,

It represents the general equation of a circle in the complex plane.

Now, let us consider a circle described on a line segment AB as its diameter. Let as its diameter. Let be any point on the circle. As the angle in the semicircle is , so

is purely imaginary.

**Condition for four points to be concyclic:**

Let ABCD be a cyclic quadrilateral such that , , and lie on a circle. (Remember the following basic property of concyclic quadrilaterals: opposite angles are supplementary).

The above property means the following:

is purely real.

Thus, points , , , (taken in order) would be concyclic if the above condition is satisfied.

More later,

Nalin Pithwa

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