The Euler line involves three interesting points associated with triangles:

• The circumcenter is the center of the circumcircle of a triangle. It is the intersection of the perpendicular bisectors of the three sides of the triangle.
• The orthocenter is the intersection of the three altitudes of the triangle.
• The centroid is the intersection of the three medians of the triangle.

Constructing the Euler Line of a Triangle (Interactively!)

For any triangle, perform the following steps. Use the interactive math tool below for this interactive construction.

• Step 1. Construct the perpendicular bisectors of the triangle.
• Step 2. Construct the circumcenter O, the intersection of three perpendicular bisectors.
• Step 3. Hide the perpendicular bisectors.
• Step 4. Construct the altitudes of the triangle.
• Step 5. Construct the orthocenter H, the intersection of the altitudes.
• Step 6. Hide the altitudes.
• Step 7. Construct the medians of the triangle
• Step 8. Construct the centroid, G, the intersection of the medians.
• Step 9. Hide the medians.
• Step 10. Construct the Euler line.

You can drag any of the vertices of the triangle to change its shape. What always happens to the circumcenter, the orthocenter, and the centroid of the triangle?

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Definition of the Euler Line

The Euler line of a triangle is the line that passes through the circumcenter, the orthocenter, and the centroid of the triangle.

• Explore the relationships among the circumcenter, the orthocenter, and the centroid of a triangle

Back to Overview of Lines

References and Credits