Math 2.0

Geometry Definition

Orthocentre

Builds on TriangleAltitude of a triangleLine

Definition

Let ABCABC be a triangle in the Euclidean space ℝn\mathbb{R}^n. The three lines that contain the altitudes of ABCABC meet in exactly one point HH. This point HH is the orthocentre of ABCABC.

Remarks

The orthocentre is inside the triangle if the triangle is acute. In a right triangle, it is the vertex at the right angle. In an obtuse triangle, it is outside. In an equilateral triangle, the orthocentre is the same point as the centroid, the circumcentre and the incentre. The Book of Lemmas, attributed to Archimedes, uses the fact that the three altitudes meet. Al-Nasawi stated and proved it explicitly in the 11th century. Mathlib's Affine.Triangle.orthocenter defines it as the Monge point of the triangle.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Altitude of a triangle, concept, no evidence levelAltitude of a triangleconceptLine, concept, no evidence levelLineconceptTriangle, concept, no evidence levelTriangleconceptOrthocentre, definition, no evidence levelOrthocentredefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Altitude of a triangle concept
  2. Line concept
  3. Triangle concept
  4. Orthocentre definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthocenter, lead section
  2. Web https://en.wikipedia.org/wiki/Orthocenter, History
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DefinesOrthocentre
UsesTriangle, Altitude of a triangle, Line

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