Every triangle has four classic centers, each the solution of a different geometric problem:
Centroid G = (A + B + C) / 3
Circumcenter O — equidistant from A, B, C
(intersection of the perpendicular bisectors)
Incenter I = (a·A + b·B + c·C) / (a + b + c)
a,b,c = side lengths opposite A,B,C
Orthocenter H — intersection of the three altitudes
Area, via the shoelace formula, and the two radii:
Area = ½ |(B−A) × (C−A)|
s = perimeter / 2
r = Area / s (incircle radius)
R = a·b·c / (4·Area) (circumcircle radius)
The most striking fact: O, G and H are always exactly collinear — the Euler line — and G divides the segment OH so that HG = 2·GO, no matter how the triangle is reshaped. Drag any vertex and watch the ratio in the box above stay pinned at 2.00 while the three points slide along one straight line.
- Drag a colored vertex sphere to reshape the triangle in real time.
- Presets jump to an equilateral, right, or obtuse triangle (in an equilateral triangle all four centers coincide).
- Circumcircle / Incircle show the unique circle through the vertices and the unique circle tangent to all three sides.