Triangles (1) - Basic terms and concepts

The TRIANGLE is of immense importance and holds many geometric mysteries.

 

*** LABELLING *** 

 

The vertices are usually labelled with upper case letters (A, B, C) and this is generally done in an anti-clockwise direction as shown in Figure 1.  The side opposite vertex A is labelled with lower case letter a and has length a units. Opposite B is side b, and opposite C is side c.

Any vertex can be chosen as the starting point and labelling is sometimes done in a clockwise direction.

For a triangle to exist, the sum of the lengths of any two sides must be greater than the length of the third side. In Figure 1, there are three INEQUALITIES:

a + b > c

a + c > b

b + c > a

 

*** TYPES of TRIANGLE ***

There are four types of triangle: Equilateral, Isosceles, Right-angled, Scalene - BBC What are the Types of Triangle?

A CEVIAN (named after Giovanni Ceva) is a straight line segment from a vertex to a point on the side opposite that vertex. MEDIANS, ANGLE BISECTORS are special cases of cevians.  

An ALTITUDE is a line segment through a vertex and perpendicular to (i.e. making a right angle with) tthe opposite side (extended if necessary).

see BBC Bitesize

*** The "Centre" of a Triangle ***

In a book published in 1998 ("Triangle Centers and Central Triangles"), Professor Clark Kimberling of the University of Evansville listed 400 triangle centres. 

On 12 March 2025, the Encyclopedia of Triangle Centers (ETC) identifies over 70,000 triangle "centres"

Encyclopedia of Triangle Centers (Wikipedia) 

Professor Clark Kimberling - Triangle Centres 

The first 4 "centres" listed are the most frequently encountered and are:

INCENTRE - the intersection of the bisectors of the interior vertex angles

CENTROID - the intersection of the medians of the triangle

CIRCUMCENTRE - the intersection of the perpendicular bisectors of the triangle's sides

ORTHOCENTRE - the intersection of the altitudes of the triangle