Triangles

Triangles are three-sided polygons that form the foundation of much geometric and trigonometric study.

Types of Triangles

By Side Length

Equilateral: All three sides equal
Isosceles: Two sides equal
Scalene: All sides different

By Angle Measure

Acute: All angles less than 90°
Right: One angle equals 90°
Obtuse: One angle greater than 90°

Triangle Properties

Fundamental Properties of Triangles

Perimeter: Sum of all sides: $P = a + b + c$ (where $a$, $b$, and $c$ are the lengths of the sides of the triangle)
Semi-perimeter: $s = \frac{a + b + c}{2}$ (used in Heron's formula–see below–for area)
Sum of angles: Always equals 180°: $α + β + γ = 180°$
Triangle inequality: Sum of any two sides > third side: $a + b > c$, $a + c > b$, $b + c > a$
Exterior angle: Equals sum of two non-adjacent interior angles: $\angle_{ext} = α + β$ (where $α$ and $β$ are the two non-adjacent interior angles)
Sum of exterior angles: Always equals 360°: $\angle_{ext1} + \angle_{ext2} + \angle_{ext3} = 360°$

Fundamental Definitions

Intersecting vs Parallel Lines: Two lines that cross each other intersect whereas parallel lines run alongside without meeting
Concurrent lines: Three or more lines that intersect at a single point called the point of concurrency. (MathWorld uses a less standard definition of "two or more" which is used in computational contexts or differential geometry.)
Collinear points: Three or more points that lie on the same straight line
Vertex: A point where two or more line segments meet; in a triangle, the three corners where sides intersect
Perpendicular (orthogonal): Two lines that intersect at a right angle (90°)
Bisect: To divide into two equal parts; a bisector divides an angle or line segment into two congruent pieces

Transversals

A transversal is a line that intersects two or more other lines. When a transversal intersects the sides of a triangle (or their extensions), it creates important geometric relationships.

Cevians

A cevian is a line segment that joins a vertex of a triangle to a point on the opposite side (or its extension).

Median: Cevian from vertex to midpoint of opposite side
Altitude: Cevian perpendicular to opposite side
Angle bisector: Cevian that bisects the angle at a vertex

Giovanni Ceva (1647-1734) was an Italian mathematician who discovered this fundamental theorem about concurrent cevians. His work laid important groundwork for projective geometry and triangle geometry.

For cevians $AA'$, $BB'$, $CC'$ in triangle $ABC$ (where $A'$, $B'$, $C'$ are on sides $BC$, $CA$, $AB$ respectively), the cevians are concurrent if and only if: $\frac{BA'}{A'C} \cdot \frac{CB'}{B'A} \cdot \frac{AC'}{C'B} = 1$

For more information, see: Ceva's Theorem - Wolfram MathWorld

Menelaus' Theorem

Menelaus of Alexandria (c. 70-140 CE) was a Greek mathematician and astronomer who developed this theorem about collinear points. His work "Sphaerica" was influential in both geometry and astronomy.

For a transversal intersecting sides $BC$, $CA$, $AB$ (or their extensions) at points $A'$, $B'$, $C'$ respectively, the points are collinear if and only if: $\frac{BA'}{A'C} \cdot \frac{CB'}{B'A} \cdot \frac{AC'}{C'B} = -1$

For more information, see: Menelaus' Theorem - Wolfram MathWorld

Circumcircle, Incircle and Special Transversals

Circumcircle: The circle that passes through all three vertices of a triangle.
Circumcenter: The point equidistant from all vertices, the center of the circumcircle.
Perpendicular bisectors: The lines passing through the midpoint of each side of a triangle and which are perpendicular to the given side. The three perpendicular bisectors intersect at the circumcenter.
Incircle: The circle that is tangent to all three sides of a triangle.
Incenter: The center of the incircle. Hence it is also the interior point for which distances to the sides of the triangle are equal. The three angle bisectors intersect at the incenter.
Centroid: The point where the three triangle medians intersect, also known as the center of mass or barycenter.
Orthocenter: The point where the altitudes of a triangle intersect.

Area Calculations

Standard Formula

For a triangle with base $b$ and height $h$: $A = \frac{1}{2}bh$

Heron's Formula

Heron's Formula is a useful way to find the area of a triangle when you only know the lengths of its sides. For a triangle with sides $a$, $b$, $c$: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter.

Using Trigonometry

If you know the length of two sides and the included angle, you can use the following trigonometric formula: For a triangle with two sides $a$, $b$ and included angle $γ$: $A = \frac{1}{2}ab \sin γ$

Coordinate Geometry

For a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$: $A = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

For more information, see: Area of Triangle - Wolfram MathWorld

Right Triangles

Fundamental Properties of Right Triangles

Right triangles have special properties and are fundamental to trigonometry.

Right Triangle: A triangle with one angle equal to 90°.
Cathetes and Hypotenuse: The two sides forming the right angle are called the cathetes or catheti, and the side opposite the right angle is called the hypotenuse.
Pythagorean Theorem: For a right triangle with legs $a$, $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$

Pythagorean Triples: a triple of positive integers a, b, and c such that a right triangle exists with legs $a$,$b$ and hypotenuse $c$. This is equivalent to finding positive integers $a$,$b$ and $c$ satisfying the Pythagorean Theorem:
- Examples:
  - 1: $(3, 4, 5)$
  - 2: $(5, 12, 13)$
  - 3: $(8, 15, 17)$
  - 4: $(7, 24, 25)$
- Primitive Pythagorean triples: These are triples where $a$, $b$, and $c$ are coprime (i.e., their greatest common divisor is 1). Examples include $(3, 4, 5)$ and $(5, 12, 13)$.
- Generating Pythagorean triples: You can generate all primitive Pythagorean triples using the formula:
  - For integers $m > n > 0$:
    - calculate $a = m^2 - n^2$
    - calculate $b = 2mn$
    - calculate $c = m^2 + n^2$
  - This generates all primitive triples, and you can multiply by a common factor to get non-primitive ones.
  - Example: For $m = 2$ and $n = 1$, you get $(3, 4, 5)$.
  - Note: The order of $a$ and $b$ does not matter, so $(4, 3, 5)$ is also a valid Pythagorean triple.
Area of Right Triangles: For a right triangle with legs $a$ and $b$: $A = \frac{1}{2}ab$

Advanced Right Triangle Theorems

Geometric Mean Theorem:
- Also known as the Altitude-on-Hypotenuse Theorem or Right Triangle Altitude Theorem.
- For right triangles, the altitude to the hypotenuse has length: $h = \sqrt{pq}$ where $p$ and $q$ are the segments of the hypotenuse.
- For a right triangle with hypotenuse $c$ divided into segments $p$ and $q$ by the altitude:
  - where $a$ is the cathetus adjacent to segment $p$: $a^2 = cp$
  - where $b$ is the cathetus adjacent to segment $q$: $b^2 = cq$
- It is called the geometric mean because:
  - From $a^2 = cp$, we get $a = \sqrt{cp}$, so $a$ is the geometric mean of $c$ and $p$.
  - From $b^2 = cq$, we get $b = \sqrt{cq}$, so $b$ is the geometric mean of $c$ and $q$.
  - Alternatively, $h = \sqrt{pq}$ is the geometric mean of the segments $p$ and $q$.
Thales' Theorem: If a triangle is inscribed in a circle such that one side is a diameter, then the triangle is a right triangle.

Special Right Triangles

Isosceles Right Triangle

Angles are 45°-45°-90°
Sides in ratio $1 : 1 : \sqrt{2}$
If legs = $a$, then hypotenuse = $a\sqrt{2}$
Derivation of the 45°-45°-90° Triangle:
- Start with an isosceles triangle where the two equal angles are each 45°.
- The hypotenuse is opposite the right angle, and the legs are equal in length.
- By the Pythagorean theorem, if each leg has length $a$, then the hypotenuse has length $a\sqrt{2}$.
Area: $A = \frac{1}{2}a^2$

30°-60°-90° Triangle

Sides in ratio $1 : \sqrt{3} : 2$
If short leg = $a$, then long leg = $a\sqrt{3}$, hypotenuse = $2a$
Derivation of the 30°-60°-90° Triangle:
- Start with an equilateral triangle with side length $2a$.
- By drawing an altitude from one vertex to the opposite side, you create two 30°-60°-90° triangles.
- The altitude bisects the base, creating a segment of length $a$.
- The altitude itself has length $a\sqrt{3}$, leading to the ratios mentioned above.

Further Information

More information about triangles are found in the trigonometry section of this documentation. Specifically, the sections on trigonometric functions provide deeper insights into the relationships between triangle angles and side lengths.

Applications

Triangulation: Determining position using angle measurements
GPS systems: Using triangular calculations

Engineering

Structural analysis: Triangular frameworks and trusses
Computer graphics: Triangular meshes for 3D modeling

Physics

Vector decomposition: Breaking forces into components
Projectile motion: Analyzing trajectories using right triangles