Properties of Triangles

Overview

Triangles are essential geometric shapes with unique properties based on their sides, angles, and structural elements. Understanding these properties helps in problem-solving and constructing geometric proofs.

Types of Triangles

By Sides:

• Equilateral Triangle: All three sides are equal; all angles are 60°.
• Isosceles Triangle: Two sides are equal; the angles opposite these sides are also equal.
• Scalene Triangle: All sides and angles are different.

By Angles:

• Acute Triangle: All angles are less than 90°.
• Obtuse Triangle: One angle is greater than 90°.
• Right Triangle: One angle is exactly 90°, with the side opposite it called the hypotenuse.

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Key Properties

1. Angle Sum Property: The sum of the angles in a triangle is always $180^\circ$.
2. Exterior Angle Property: An exterior angle is equal to the sum of the two non-adjacent interior angles.
3. Inequality Property: The sum of the lengths of any two sides of a triangle is greater than the third side (Triangle Inequality Theorem).
4. Pythagoras' Theorem (Right Triangles): In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
5. Special Points:

• Centroid: Intersection of medians, dividing them in a 2:1 ratio.
• Orthocenter: Intersection of altitudes.
• Circumcenter: Intersection of perpendicular bisectors, the centre of a circle passing through all vertices.
• Incenter: Intersection of angle bisectors, the centre of the inscribed circle.

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Worked Examples

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Summary

• Types of Triangles: Categorized by sides (equilateral, isosceles, scalene) and angles (acute, obtuse, right).
• Key Properties:
  • Angle sum is always 180°
  • The exterior angle equals the sum of two non-adjacent interior angles.
  • Satisfies the triangle inequality property.
  • Special points include the centroid, orthocenter, circumcenter, and incenter.
• Applications: Used in constructions, trigonometry, and proofs.