Triangle Inequality Calculator

Triangle Existence: -

Side 3 Range: -

Understanding Triangle Inequality

What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the third side, and the absolute difference of any two sides must be less than the third side.

For sides a, b, and c:

  • a + b > c
  • b + c > a
  • c + a > b
  • |a - b| < c
  • |b - c| < a
  • |c - a| < b

Advanced Triangle Properties

  • Isosceles Triangle: Two sides equal
  • Equilateral Triangle: All sides equal
  • Scalene Triangle: No sides equal
  • Right Triangle: Follows Pythagorean theorem
  • Triangle Area: √(s(s-a)(s-b)(s-c)) where s=(a+b+c)/2
  • Perimeter: Sum of all sides
  • Semiperimeter: Half of perimeter
  • Inradius: Area/Semiperimeter

Geometric Implications

Medians

Lines from vertices to midpoints of opposite sides

Altitudes

Perpendicular lines from vertices to opposite sides

Angle Bisectors

Lines that divide angles into equal parts

Circumcenter

Intersection of perpendicular bisectors

Applications in Mathematics

  • Euclidean Geometry: Fundamental theorem
  • Vector Analysis: Triangle inequality in vector spaces
  • Optimization: Shortest path problems
  • Network Theory: Triangle routing
  • Computational Geometry: Mesh generation
  • Physics: Force diagrams and equilibrium
  • Engineering: Structural analysis
  • Computer Graphics: Polygon mesh validation