The Center and Radius of the Nine-Point Circle

circumcircle radii
nine-point circle radii

R

≈

0.92

r

≈

0.46

2

×

r

≈

0.92

ON

≈

0.25

NH

≈

0.25

Let ABC be a triangle with orthocenter H, circumcenter O, and circumradius

R

. Let the nine-point circle have center N and radius

r

.

Then N lies on OH (the Euler line) and bisects OH.

Also,

R=2r

.

See theorem 209 in N. Altshiller-Court, College Geometry, Mineola, NY: Dover, 2007 p. 104.

Jay Warendorff