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What Is the Euler Line?

Euler line of a triangle

The illustration above shows an important phenomenon in geometry—namely that the orthocenter, centroid, and the circumcenter of a triangle lie on the same line. The line is called the Euler line.

Theory

The Euler Line

The Euler line is the straight line passing through the orthocenter, centroid, and circumcenter of a triangle.

Example 1

Given the points $(1, 2)$ , $(- 1, 3)$ and $(2, 0)$ , can these points be the orthocenter, centroid and circumcenter of a triangle?

To find out, you use the point-slope equation

y - y_{1} = a (x - x_{1})

on two of the points and check if the third is on the line. First, find the slope $a$ :

a = \frac{Δ y}{Δ x} = \frac{2 - 3}{1 - (- 1)} = - \frac{1}{2}

Now use one of the points you used to calculate the slope, and add it to the formula. Then you get:

\begin{array}{l} y - 2 & = - \frac{1}{2} (x - 1) \\ y & = - \frac{1}{2} x + \frac{1}{2} + 2 \\ = - \frac{1}{2} x + \frac{5}{2} \end{array}

Now put the last point $(2, 0)$ in the expression $y$ and see if it fits:

0 \neq \frac{3}{2} = - \frac{1}{2} \cdot 2 + \frac{5}{2}

You can thereby conclude that the three points cannot be the orthocenter, centroid and circumcenter of a triangle. If this was the case, they would all lie on the Euler line.

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