Lesson 1 — Triangles and Circles

Big idea: Every triangle has a circle through all three vertices, and its center is where the perpendicular bisectors of the sides meet.

Words to know

Word	Meaning
Cyclic	A polygon is cyclic iff there is a circle that contains all of its vertices.
Circumcircle	The circle that passes through all three vertices of a triangle.
Circumcenter	The center of the circumcircle; where the perpendicular bisectors of the sides meet.
Inscribed in a circle	A polygon is inscribed in a circle iff each vertex lies on the circle.
Circumscribed about a polygon	A circle is circumscribed about a polygon iff it contains all of the polygon’s vertices.
Perpendicular bisector	A line perpendicular to a segment at its midpoint.
Concurrent	Three or more lines are concurrent iff they contain the same point.

Theorem 68: Every triangle is cyclic.
Proof idea: the perpendicular bisectors of two sides meet at P, and P is equidistant from all three vertices; so a circle centered at P passes through A, B, and C.
Corollary to Theorem 68: The perpendicular bisectors of the sides of a triangle are concurrent.
The circumcenter is equidistant from the three vertices.
Three noncollinear points determine a circle; two points are not enough.
The circumcenter is inside an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle.
Construction 9: To circumscribe a circle about a triangle, construct the perpendicular bisectors of two sides; their intersection is the center, and the radius is the distance to any vertex.

No new formula. The radius is OA = OB = OC.
Two perpendicular bisectors are enough; the third must pass through the same point.
For a right triangle, the circumcenter is the midpoint of the hypotenuse, so the circumradius is half the hypotenuse.

If the homework asks…	Do this…
Find the circumcenter	Construct the perpendicular bisectors of two sides.
Find the circumradius	Measure from the circumcenter to any vertex.
Check whether a point is the circumcenter	Check that its distances to all three vertices are equal.
Work with a right triangle	Use the midpoint of the hypotenuse as the circumcenter.
Prove three lines are concurrent	Show that all three contain the circumcenter.
Draw a circle through three points	Connect the points and bisect two of the segments.

The circumcenter is not always inside the triangle.
A perpendicular bisector is not a median or an altitude.
Two points determine many circles; you need three noncollinear points.
“Circle circumscribed about a polygon” and “polygon inscribed in a circle” describe the same picture from opposite viewpoints.