1. Triangle Notation and Terminology

In triangle geometry, understanding notation is the first step:

• Capital letters \( A, B, C \) refer to angles or vertices.
• Small letters \( a, b, c \) represent the sides opposite those angles:
  • \( a = BC \)
  • \( b = CA \)
  • \( c = AB \)
• A line drawn from a vertex to the opposite side is called a Cevian (e.g., \( AD \)).
• A line drawn from one side to another side (not passing through a vertex) is called a Side Cut (e.g., \( PQ \)).

2. Important Cevians in a Triangle

There are three well-known cevians in triangle geometry, each associated with a special point:

• Median: Connects a vertex to the midpoint of the opposite side. All medians intersect at the centroid (G).
• Altitude: A perpendicular from a vertex to the opposite side. All altitudes meet at the orthocenter (O).
• Angle Bisector: Divides the angle into two equal parts. All angle bisectors intersect at the incenter (I).

3. Triangle Inequality Theorem

To form a valid triangle, the sum of the lengths of any two sides must be greater than the third side.

Let’s analyze:

• If \( AB + BC < CA \): Cannot form a triangle – it’s an open figure.
• If \( AB + BC = CA \): Points are collinear – they lie on a straight line.
• If \( AB + BC > CA \): A valid triangle is formed.

4. Alternate View of Triangle Side Conditions

Consider ordering the sides such that:

Now observe:

• If \( AB + BC < CA \): Triangle cannot be formed (sum of small sides < largest side).
• If \( AB + BC = CA \): Points are collinear (they form a straight line).
• If \( AB + BC > CA \): Triangle exists (sum of smaller sides is more than the largest side).

Summary

• Use uppercase letters for angles/vertices and lowercase for sides.
• Cevians are lines from vertex to opposite side – medians, altitudes, and angle bisectors are special cevians.
• Always check the triangle inequality: The sum of any two sides must be greater than the third side.

This is a foundational concept that appears across geometry, trigonometry, and even in competitive exams like CAT, SSC, and JEE.