Math 2.0

Geometry Theorem

Isosceles triangle theorem

Builds on TriangleAngle at a pointEuclidean distance

Statement

Let ABCABC be a triangle in a Euclidean space ℝn\mathbb{R}^n with d(A,B)=d(A,C)d(A, B) = d(A, C), where dd is the Euclidean distance. Then the angles at BB and CC are equal: ∠ABC=∠ACB\angle ABC = \angle ACB.

Remarks

The converse holds too: if ∠ABC=∠ACB\angle ABC = \angle ACB in a triangle ABCABC, then d(A,B)=d(A,C)d(A, B) = d(A, C). Mathlib's form of the statement does not need that AA, BB and CC are not collinear. In an equilateral triangle, the theorem applies at each vertex. So all three angles are equal.

History

The statement is Proposition 5 of Book I of Euclid's Elements. Its Latin name is pons asinorum, "the bridge of donkeys".

Reading path

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

Show the path as a graph
Angle at a point, concept, no evidence levelAngle at a pointconceptEuclidean distance, concept, no evidence levelEuclidean distanceconceptTriangle, concept, no evidence levelTriangleconceptIsosceles triangle theorem, statement, no evidence levelIsosceles triangle th…statement
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. Angle at a point concept
  2. Euclidean distance concept
  3. Triangle concept
  4. Isosceles triangle theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Isosceles_triangle, Terminology, classification, and examples
About this page
Tag056VS31A0J
URIm2://056VS31A0J
Aliasespons-asinorum, base-angles-theorem
Kindstatement, theorem
WritingW0 stub
MSC51M04
StatusThis copy of the site has no build data. Run m2 status 056VS31A0J.
UsesTriangle, Angle at a point, Euclidean distance

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