Arc of a Circle

Locus of z from which the segment z₁z₂ subtends a constant angle α. This is an arc of a circle passing through z₁ and z₂ — the inscribed angle theorem in complex language. The full circle is obtained by taking both α and π−α. The complementary arc gives angle π−α.

The locus $\arg\!\left(\dfrac{z - z_1}{z - z_2}\right) = \alpha$ is the complex form of the inscribed angle theorem — one of the most elegant connections between complex numbers and classical geometry.

The Angle Subtended by a Chord

Let $P(z)$ be a point and $z_1$, $z_2$ be fixed. The angle at $P$ subtended by the chord $z_1 z_2$ is the angle $\angle z_1 P z_2$.

The vector from $P$ to $z_1$ is $z_1 - z$, and from $P$ to $z_2$ is $z_2 - z$. The angle between them:

$$\angle z_1 P z_2 = \arg\!\left(\frac{z_1 - z}{z_2 - z}\right) = \arg\!\left(\frac{z - z_1}{z - z_2}\right)$$

(The sign of the ratio changes by $(-1)/(-1) = 1$, so the argument is the same.)

The Inscribed Angle Theorem

The inscribed angle theorem states: all points on the same arc of a circle subtend equal angles to the chord.

So the locus $\arg\!\left(\dfrac{z - z_1}{z - z_2}\right) = \alpha$ is precisely the arc of the circle through $z_1$ and $z_2$ from which the chord subtends angle $\alpha$.

The Full Circle

The complementary arc (on the other side of $z_1 z_2$) subtends angle $\pi - \alpha$ at every point. Together, $\alpha$ and $\pi - \alpha$ cover the full circle:

$$\arg\!\left(\frac{z - z_1}{z - z_2}\right) = \alpha \quad \cup \quad \arg\!\left(\frac{z - z_1}{z - z_2}\right) = \alpha - \pi$$

gives both arcs and hence the complete circle (excluding $z_1$ and $z_2$).

Special Case — Semicircle ($\alpha = \pi/2$)

When $\alpha = \pi/2$, the chord $z_1 z_2$ is a diameter and the arc is a semicircle. Every point on it sees $z_1 z_2$ at $90°$ — this is Thales' theorem:

$$\arg\!\left(\frac{z - z_1}{z - z_2}\right) = \frac{\pi}{2} \iff \frac{z - z_1}{z - z_2} \text{ is purely imaginary}$$

Degenerate Cases

• $\alpha = 0$ or $\pi$: the ratio is real — $z$ lies on the line through $z_1$ and $z_2$ (the arc degenerates to the line).
• As $\alpha \to 0$, the arc grows larger; as $\alpha \to \pi/2$, the arc shrinks toward the semicircle.