Diameter Form

Circle with (x₁, y₁) and (x₂, y₂) as endpoints of a diameter. Follows directly from the angle-in-semicircle being 90°.

Let $A(x_1, y_1)$ and $B(x_2, y_2)$ be the endpoints of a diameter, and $P(x, y)$ any other point on the circle.

By the angle-in-semicircle theorem, $\angle APB = 90°$. Therefore $\overrightarrow{PA} \perp \overrightarrow{PB}$:

$$\overrightarrow{PA} \cdot \overrightarrow{PB} = 0$$ $$(x_1 - x)(x_2 - x) + (y_1 - y)(y_2 - y) = 0$$

Rewriting:

$$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$$

This is the diameter form. It is particularly useful when the endpoints of a diameter are given directly, avoiding the need to find the centre and radius separately.