Points, Lines, 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)

Exercises

Exercise 1

Find the lengths of the three altitudes, h_a, h_b and h_c, of the triangle Δ ABC, if you know the lengths of the three sides: a=3 cm, b=4 cm and c=4.5 cm.

Firstly, we calculate the semiperimeter (s).

We get that semiperimeter is s = 5.75 cm. Then we can find the altitudes:

The lengths of three altitudes will be h_a=3.92 cm, h_b=2.94 cm and h_c=2.61 cm.

Exercise 2

Find the length of the median of a triangle Δ ABC if length of sides are a=2 cm, b=4 cm and c=3 cm.

Using the equation of Apollonius’ theorem, we can find the length of all three medians:

Thus, the medians are m_a=3.39 cm, m_b=1.58 cm and m_c=2.78 cm.

Exercise 3

Find the radius R of the circumscribed circle (or circumcircle) of a triangle of sides a = 9 cm, b = 7  cm and c = 6 cm.

Solution:

We get the semiperimeter s:

We apply the formula for the radius R of the circumscribed circle, giving the following values:

So, the radius R is 4.5 cm.

Exercise 4

Let ABC de a triangle in which a=3 cm, b=4 cm and c=2 cm. Find the angle bisectors B_a, B_b y B_c.

SOLUTION:

We’re going to solve this exercise using the angle bisector’s formula:

Since we already know what all the sides of the triangle measure, we only have to find the semiperimeter (s):

Then, we can substitute the values in the angle bisector’s formula:

Therefore, B_a=2.45 cm, B_b=1.47 cm and B_c=3.32 cm.