Home

Nine-Point Circle

Nine-point circle calculator. Enter three vertices to compute the nine-point center N and radius r, with circumcenter relation.

展开更多 ▾
N=(O+H)/2, r=R/2

📖 Tutorial | Nine-Point Circle

1. Definition
Nine-point circle passes through 9 points: side midpoints, altitude feet, and midpoints from vertices to the orthocenter. Its center N is the midpoint of O and H; its radius is half the circumradius R.
2. Symbols
SymbolMeaning
Ocircumcenter
Horthocenter
Nnine-point center=(O+H)/2
Rcircumradius; nine-point radius=R/2
3. How it works
  • Find the circumcenter O;
  • Find H=A+B+C−2O;
  • Nine-point center N=(O+H)/2;
  • Nine-point radius=R/2.
4. Steps
  1. Enter three vertex coordinates;
  2. Click Calculate;
  3. Read O, H, N and the radius;
  4. Check N is the midpoint of O and H.
5. Example
Example: right triangle (0,0)(4,0)(0,3). O=(2,1.5), H=(0,0), N=(1,0.75), R=2.5, nine-point radius=1.25.
6. Pitfalls
The three points must not be collinear;
N is always the midpoint of O and H;
The nine-point radius is always R/2.

❓ FAQ | Nine-Point Circle

Which nine points?
Side midpoints, altitude feet, and midpoints from vertices to H.
Where is the nine-point center?
At the midpoint of O and H, on the Euler line.
What is its radius?
Half the circumradius R.
Special for a right triangle?
The right-angle vertex is H, and O is the hypotenuse midpoint.
Relation to the centroid?
N lies on the Euler line, but it is the midpoint of O,H, not the centroid.