Euler Line Calculator
Free Euler line Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. Enter your values for instant results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
H = 3G - 2O, OG:GH = 1:2
The orthocenter H, centroid G, and circumcenter O are collinear (Euler line). The centroid divides segment OH in ratio 1:2 from O. The nine-point center N is the midpoint of OH with radius R/2.
Worked Examples
Example 1: Euler Line of a Scalene Triangle
Problem:Find the Euler line for a triangle with vertices A(0,0), B(8,0), C(3,6).
Solution:Centroid G = ((0+8+3)/3, (0+0+6)/3) = (3.6667, 2)\nCircumcenter O: Using perpendicular bisector formulas = (4.4167, 2.2917)\nOrthocenter H = 3G - 2O = (3x3.6667 - 2x4.4167, 3x2 - 2x2.2917) = (2.1667, 1.4167)\nNine-point center N = ((4.4167+2.1667)/2, (2.2917+1.4167)/2) = (3.2917, 1.8542)\nOG:GH ratio = 1:2 confirmed
Result:G(3.6667, 2) | O(4.4167, 2.2917) | H(2.1667, 1.4167) | Euler line verified
Example 2: Euler Line of a Right Triangle
Problem:Find the Euler line for a right triangle with vertices A(0,0), B(6,0), C(0,8).
Solution:Centroid G = (2, 2.6667)\nCircumcenter O = midpoint of hypotenuse = (3, 4)\nOrthocenter H = right angle vertex = (0, 0)\nNine-point center N = (1.5, 2)\nEuler line slope = (4-0)/(3-0) = 4/3\nEuler line: y = (4/3)x\nOG = sqrt(1+1.7778) = 1.6667, GH = sqrt(4+7.1111) = 3.3333, ratio = 1:2
Result:O(3,4) | G(2, 2.667) | H(0,0) | Euler line: y = 1.3333x
Frequently Asked Questions
What is the Euler line of a triangle?
The Euler line is a remarkable straight line that passes through several important centers of a non-equilateral triangle. Named after the great mathematician Leonhard Euler who proved its existence in 1765, this line passes through the circumcenter (center of the circumscribed circle), the centroid (center of mass), and the orthocenter (intersection of altitudes). The nine-point center also lies on this line. For equilateral triangles, all these centers coincide at a single point, so the Euler line is undefined. The discovery of the Euler line was a major milestone in triangle geometry.
What points lie on the Euler line?
The Euler line passes through four major triangle centers. The circumcenter O is the center of the circle passing through all three vertices. The centroid G is the intersection of the three medians and the center of mass. The orthocenter H is where the three altitudes meet. The nine-point center N is the center of the circle passing through the midpoints of the sides, the feet of the altitudes, and the midpoints of segments from vertices to the orthocenter. Notably, the incenter (center of the inscribed circle) generally does NOT lie on the Euler line, except for isosceles triangles.
What is the ratio of distances along the Euler line?
The points on the Euler line maintain fixed distance ratios. The centroid G divides the segment from the circumcenter O to the orthocenter H in the ratio OG:GH = 1:2. This means the orthocenter is always twice as far from the centroid as the circumcenter is. The nine-point center N is the midpoint of the segment OH, so ON = NH = OH/2. Also, NG = OH/6. These ratios hold for every non-equilateral triangle regardless of its shape or size. This beautiful property was first proved by Euler and is one of the most elegant results in classical geometry.
Does the incenter lie on the Euler line?
In general, the incenter (center of the inscribed circle) does NOT lie on the Euler line. The incenter only coincides with points on the Euler line in special cases. For equilateral triangles, all centers coincide, so the incenter trivially lies on the (degenerate) Euler line. For isosceles triangles, the Euler line is the axis of symmetry, and the incenter also lies on this axis, so it does lie on the Euler line. For all other (scalene, non-equilateral) triangles, the incenter is not on the Euler line. This makes the incenter unique among the major triangle centers.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy