Geometry • Topic 12

Euler Line

The Euler Line is a beautiful line that connects several important triangle centers.

Theorem

Theorem (Euler Line)

In any triangle, the Centroid (

G

), Circumcenter (

O

), and Orthocenter (

H

) are collinear. Furthermore,

G

lies between

O

and

H

with the ratio:

H G : GO = 2 : 1

Vector Proof

Let the origin be the circumcenter

O

. Then

G = \frac{1}{3} (A + B + C)

. It is a known fact (Hamilton's Theorem) that

H = A + B + C

. Thus,

H = 3 G

, which implies

O, G, H

are collinear and

O H = 3 OG

Center of the Nine-Point Circle ( $N$ )

The center of the Nine-Point Circle also lies on the Euler Line. It is the midpoint of

O H

. Order of points:

O - G - N - H

Special Case

In an equilateral triangle,

O, G, H, I

all coincide. The Euler line is undefined (or technically, any line through this point). In an isosceles triangle, the Euler line coincides with the axis of symmetry.

Practice Problems

Exercise (Problem 1)

△ A BC

O, G, H

are defined as usual. If the distance

O H = 12

, find the distance

G H

Exercise (Problem 2)

Prove that in a right triangle, the orthocenter is the vertex at the right angle, and the circumcenter is the midpoint of the hypotenuse. Verify the

2 : 1

ratio for the centroid.

Exercise (Problem 3)

If the Euler line passes through the vertex

A

, prove that the triangle is either isosceles (

A B = A C

) or right-angled at

A