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Euler Line Explorer

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About This MicroSim

This interactive visualization demonstrates the Euler line - one of the most remarkable discoveries in triangle geometry. Named after Leonhard Euler (1707-1783), this line passes through three important triangle centers.

Key Concepts

The Euler Line

In any non-equilateral triangle, the following three points are collinear (lie on the same line): - G - Centroid (intersection of medians) - O - Circumcenter (intersection of perpendicular bisectors) - H - Orthocenter (intersection of altitudes)

The 2:1 Ratio

The centroid G divides the segment from O to H in the ratio 1:2: - Distance OG : Distance GH = 1 : 2 - The centroid is located 1/3 of the way from O to H

Special Cases

  • Equilateral triangle: All four centers (including incenter) coincide at a single point
  • The incenter generally does NOT lie on the Euler line

Interactions

  • Drag vertices A, B, or C to reshape the triangle
  • Watch the three centers move along the Euler line
  • Verify the 1:2 ratio labels between O, G, and H

Learning Objectives

  • Understand that G, O, and H are always collinear (except equilateral)
  • Apply the 2:1 ratio property of the Euler line
  • Recognize how center positions change while maintaining collinearity
  • Appreciate the elegant mathematical relationship

Bloom's Taxonomy Level

Analyzing and Evaluating