Euler Line Calculator
Calculate the Euler line of any triangle from vertex coordinates. Find the circumcenter, centroid, orthocenter, nine-point center, and the Euler line
Euler Line Calculator
Calculator
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Formula: H = 3G - 2O, OG:GH = 1:2
Worked example — G(3.6667, 2) | O(4, 1.75) | H(3, 2.5) | N(3.5, 2.125) | Euler line verified
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) Circumcenter O: the perpendicular bisector of AB is x = 4; the perpendicular bisector of AC passes through (1.5, 3) with slope -1/2, giving y = 3 - 0.5(4 - 1.5) = 1.75 So O = (4, 1.75). Check: OA^2 = OB^2 = OC^2 = 19.0625. Orthocenter H = 3G - 2O = (3x3.6667 - 2x4, 3x2 - 2x1.75) = (3, 2.5) Check: the altitude from C is the vertical line x = 3, and the altitude from A has slope 5/6, giving y = 2.5 at x = 3. Nine-point center N = midpoint of OH = ((4+3)/2, (1.75+2.5)/2) = (3.5, 2.125) OG:GH ratio = 1:2 confirmed
Result:G(3.6667, 2) | O(4, 1.75) | H(3, 2.5) | N(3.5, 2.125) | 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) Circumcenter O = midpoint of hypotenuse = (3, 4) Orthocenter H = right angle vertex = (0, 0) Nine-point center N = (1.5, 2) Euler line slope = (4-0)/(3-0) = 4/3 Euler line: y = (4/3)x OG = 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.
How do you find the orthocenter of a triangle?
The orthocenter is the intersection point of the three altitudes of a triangle. An altitude is a perpendicular line from a vertex to the opposite side (or its extension). Given the circumcenter O and centroid G, the orthocenter H can be computed using the relationship H = 3G - 2O, since the centroid divides the Euler line segment OH in ratio 1:2. Alternatively, you can find two altitudes algebraically using point-slope equations with perpendicular slopes and solve the system. For acute triangles the orthocenter is inside, for right triangles at the right-angle vertex, for obtuse triangles outside.
What is the nine-point circle and its center?
The nine-point circle is a remarkable circle that passes through nine significant points of a triangle: the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from vertices to the orthocenter. Its center N lies on the Euler line at the midpoint between the circumcenter O and orthocenter H. Its radius equals exactly half the circumradius. The nine-point circle was discovered independently by Charles Brianchon, Jean-Victor Poncelet, and Karl Feuerbach in the early 19th century. Feuerbach also proved that it is tangent to the incircle and all three excircles.
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.
What happens to the Euler line for special triangles?
The Euler line behaves differently for special triangle types. For equilateral triangles, all centers merge into one point, so the Euler line is undefined (or considered a single point). For isosceles triangles, the Euler line coincides with the axis of symmetry (the perpendicular bisector of the base). For right triangles, the circumcenter lies on the hypotenuse midpoint, the orthocenter is at the right-angle vertex, and the Euler line connects them through the centroid. As a triangle becomes increasingly obtuse or elongated, the Euler line extends further as the orthocenter and circumcenter move farther apart.
How do you calculate the equation of the Euler line?
To find the equation of the Euler line, you need at least two of its known points (circumcenter, centroid, orthocenter, or nine-point center). Once you have two points, say the circumcenter O(Ox, Oy) and the centroid G(Gx, Gy), compute the slope m = (Gy - Oy) / (Gx - Ox). Then the line equation is y - Oy = m(x - Ox), or equivalently y = mx + b where b = Oy - m times Ox. If the Euler line is vertical (Gx = Ox), the equation is simply x = Ox. You can verify your result by checking that the orthocenter also satisfies this equation.
What is the historical significance of the Euler line?
The Euler line, discovered by Leonhard Euler in 1765, was one of the first major results in modern triangle geometry and demonstrated that seemingly unrelated triangle centers are connected by a deep geometric relationship. Before Euler, the circumcenter, centroid, and orthocenter were studied independently. His discovery that these three points are always collinear (except for equilateral triangles) opened up the field of triangle geometry and inspired centuries of further research. The nine-point circle was discovered later by Feuerbach, Brianchon, and Poncelet, adding another layer of beauty to Euler line geometry.
Why is the centroid a plain average of the vertices but the incenter is not?
Both are barycentric combinations, but with different weights. The centroid weights all three vertices equally, so its coordinates are the arithmetic mean of the vertex coordinates. The incenter weights each vertex by the length of the side opposite it, so its x-coordinate is a times Ax plus b times Bx plus c times Cx, all divided by the perimeter. Those side-dependent weights are what pull the incenter off the Euler line for scalene triangles, and they collapse to equal weights only when the triangle is equilateral.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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