Circumcircle of a triangle

A circle passing through all three vertices of a triangle is its circumcircle, and its center is the circumcenter¹. We find them for the triangle with vertices $A(-4, 0)$ , $B(4, 0)$ and $C(0, 4)$ .

How the circumcenter is built

The circumcenter is the intersection of the perpendicular bisectors of the three sides. A point on the perpendicular bisector of a side is equidistant from its two endpoints. The intersection of two of them is therefore equidistant from all three vertices, and the third must pass through it as well.

Side	Midpoint	Perpendicular bisector
$AB$	$(0, 0)$	$x = 0$
$AC$	$(-2, 2)$	$y = -x$

The two meet at $(0, 0)$ , which is the circumcenter. The radius is its distance to a vertex, $R = 4$ , and the circumcircle is $x^2 + y^2 = 16$ . Substituting each of the three vertices gives $16$ every time, confirming that they lie on one circle.

It is a right triangle

Since $\overrightarrow{CA} \cdot \overrightarrow{CB} = (-4)(4) + (-4)(-4) = 0$ , the right angle is at $C$ . For a right triangle the circumcenter falls at the midpoint of the hypotenuse and the radius is half of it. Conversely, the angle subtended by a diameter from a point on the circle is always a right angle: that is Thales's theorem².

Checking the radius another way

The law of sines gives it too. Here $a = BC = 4\sqrt{2}$ and $\angle A = 45^\circ$ .

\frac{a}{\sin A} = \frac{4\sqrt{2}}{\frac{\sqrt{2}}{2}} = 8 = 2R

The formula from the sides and the area agrees. With $b = CA = 4\sqrt{2}$ , $c = AB = 8$ and $S = 16$ we get the following.

R = \frac{abc}{4S} = \frac{4\sqrt{2} \times 4\sqrt{2} \times 8}{4 \times 16} = \frac{256}{64} = 4

Method	Formula	Result
Distance to a vertex	$\sqrt{0^2 + 4^2}$	$4$
Law of sines	$\dfrac{a}{2\sin A}$	$4$
Sides and area	$\dfrac{abc}{4S}$	$4$

When no circumcircle exists

If the three points lie on one line there is no circumcircle: two of the perpendicular bisectors become parallel and never meet. As long as the triangle really is a triangle, its circumcircle is uniquely determined.

Where the circumcenter sits

Shape of the triangle	Position of the circumcenter
Acute	inside
Right	the midpoint of the hypotenuse
Obtuse	outside

The incenter is always inside, but the circumcenter is not.

The three lines on the graph are the three sides, the two arcs are the circumcircle, and the large dots are the three vertices together with the circumcenter $(0, 0)$ .

Circumcircle, Wikipedia
Thales's theorem, Wikipedia