Basic Elements of a Triangle

The basic elements of a triangle are:

• Vertices: points where two sides meet. It has 3 vertices (A, B, and C).
• Sides: line segments that join two consecutive vertices of the triangle and that delimit its perimeter. It has 3 sides (a, b, and c).
• Interior angles: angles that form two consecutive sides at the vertex where they converge. There are 3 interior angles (α, β and γ). The interior angles of the triangle add up to 180° (why do they add up to 180°?):
  
• Exterior angles: angle of one side with the exterior extension of the consecutive side. There are 3 exterior angles (θ). The outside angles always add up to 360°.
• Altitude of a Triangle: the altitude (or height) of a triangle (h) is a line segment perpendicular to a side that goes from the opposite vertex to this side (or its extension). It can also be understood as the distance from one side to the opposite vertex. A triangle has three heights, depending on the reference vertex that is chosen. The three altitudes intersect at a point called the orthocenter.

Basic Elements of a Right Triangle

In a right triangle, different elements can be distinguished, referring to its sides and angles.

• Legs (or cathetus): are the sides of the triangle that together form the right angle.
• Hypotenuse: is the largest side of the triangle opposite the right angle.
• Right angle: is a 90° angle formed by the two legs.
• Acute angles: the other two angles of the triangle (α and β) are less than 90°.

Points, Lines (or Cevians), and Circles Associated with a Triangle

Every triangle has lines (also called cevians) and points that determine a number of important elements.

There are many different constructions that find a special point associated with a triangle, satisfying some unique property. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point.

Altitude of a Triangle

The altitude of a triangle, or height, is a line from a vertex to the opposite side, that is perpendicular to that side. It can also be understood as the distance from one side to the opposite vertex.

Every triangle has three altitudes (h_a, h_b and h_c), each one associated with one of its three sides. If we know the three sides (a, b, and c) it’s easy to find the three altitudes, using the Heron’s formula:

The three altitudes of a triangle (or its extensions) intersect at a point called orthocenter.

The altitude can be inside the triangle, outside it, or even coincide with one of its sides, it depends on the type of triangle it is:

• Obtuse triangle: The altitude related to the longest side is inside the triangle (see h_c, in the triangle above) the other two heights are outside the triangle (h_a, and h_b).
• Right triangle: The altitude with respect to the hypotenuse is interior, and the other two altitudes coincide with the legs of the triangle.
• Acute triangle: all three altitudes lie inside the triangle.

Where is the orthocenter located?

• Obtuse triangle: the orthocenter is outside the triangle.
• Right triangle: the orthocenter coincides with the right angle’s vertex.
• Acute triangle: the orthocenter is an inner point.

Median of a Triangle

The median of a triangle is a line segment that joins one of its vertices with the center of the opposite side.

Every triangle has three medians (m_a, m_b and m_c), one from each vertex. The lengths of the medians can be obtained from Apollonius’ theorem as:

Where a, b, and c are the sides of the triangle with respective medians m_a, m_b and m_c from their midpoints.

A triangle‘s three medians are always concurrent. The point where the medians intersect is the barycenter or centroid (G).

In any median of a triangle, the distance between the center of gravity (or centroid) G and the center of its corresponding side is one third (1/3) of the length of that median, i.e., the centroid is two thirds (2/3) of the distance from each vertex to the midpoint of the opposite side.

Each median divides the triangle into two triangles with equal areas.

Indeed, the two triangles Δ ABP and Δ PBC have the same base. AP = PC, by the same definition of the median, and the same altitude h referred to that line of the two bases from the vertex B.

In physics, the barycenter, or centroid (G), would be the center of gravity of the triangle.

Perpendicular Bisector of a Triangle

The perpendicular bisector of a segment is a line perpendicular to the segment that passes through its midpoint. It has the property that each of its points is equidistant from the segment’s endpoints.

A perpendicular bisector of a triangle ABC is a line passing through the midpoint M of each side which is perpendicular to the given side. For example, the perpendicular bisector of side a is M_a.

There are three perpendicular bisectors in a triangle: M_a, M_b and M_c. Each one related to its corresponding side: a, b, and c.

These three perpendicular bisectors of the sides of a triangle meet in a single point, called the circumcenter.

The circumcenter is the center of the triangle’s circumscribed circle, or circumcircle, since it is equidistant from its three vertices.

The radius (R) of the circumcircle is given by the formula:

The relationship between the radius R of the circumcircle, whose center is the circumcenter O, and the inradius r, whose center is the incenter I, can be expressed as follows:

Angle Bisector of a Triangle

The angle bisector of a triangle is a line segment that bisects one of the vertex angles of a triangle, and ends up on the corresponding opposite side.

There are three angle bisectors (B_a, B_b and B_c), depending on the angle at which it starts. We can find the length of the angle bisector by using this formula:

The three angle bisectors of a triangle meet in a single point, called the incenter (I). This point is always inside the triangle.

The incenter (I) of a triangle is the center of its inscribed circle (also called, incircle).

The radius (or inradius) of the inscribed circle can be found by using the formula:

Summary

The main points and lines (cevian) associated with a triangle are summarized in the following list::

• Median………….. Centroid (G)
• Altitude……………… Orthocenter (H)
• Perpendicular Bisector……….. Circumcenter (O)
• Angle Bisector…………. Incenter (I)