Points, Lines, and Circles Associated with a Triangle

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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

Drawing of the three altitudes of a triangle and the orthocenter

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 (ha, hb and hc), 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:

Formula of the three heights of the triangle

Drawing of the three heights of a triangle and the orthocenter

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:

Where is the orthocenter located?

Download this calculator to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.

Triangle-total.rar         or   Triangle-total.exe      

Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.

Median of a Triangle

Drawing of the three medians of a triangle and the centroid

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

Formula of the segments in the three medians of the triangle

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

Formula of the three medians of the triangle

Where a, b, and c are the sides of the triangle with respective medians ma, mb and mc 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.

Drawing of the division of a triangle in two by a median

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

Drawing of the three perpendicular bisectors of a triangle and the circumcenter

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 Ma.

Drawing of the three bisectors of a triangle, the circumcenter and the circumscribed circumference

There are three perpendicular bisectors in a triangle: Ma, Mb and Mc. 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:

Formula for the radius of the circumscribed circle in the triangle

Drawing of circumscribed and inscribed circles for the relationship between their radius

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:

Formula for the relationship between the radius of the circumscribed and inscribed circle in a triangle

Angle Bisector of a Triangle

Drawing of the three angle bisectors of a triangle and the incenter

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 (Ba, Bb and Bc), depending on the angle at which it starts. We can find the length of the angle bisector by using this formula:

Formula of the three angle bisectors of the triangle

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

Drawing of the three angle bisectors of a triangle, the incenter and the inscribed circle

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:

Formula for the inradius inscribed in the triangle with center at the incenter

Summary

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

Exercises

Exercise 1

Drawing in exercise 1

Find the lengths of the three altitudes, ha, hb and hc, 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).

Calculation of the semiperimeter of a triangle in exercise 1

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

Solution in exercise 1

The lengths of three altitudes will be ha=3.92 cm, hb=2.94 cm and hc=2.61 cm.

Exercise 2

Drawing of the exercise 1

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:

Solution of the exercise 1

Thus, the medians are ma=3.39 cm, mb=1.58 cm and mc=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.

Drawing of the exercise 1 of the circumcenter of a triangle

Solution:

We get the semiperimeter s:

Calculation of the semiperimeter in exercise 1

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

Calculation of the radius in exercise 1

So, the radius R is 4.5 cm.

Drawing of the result of exercise 1

Exercise 4

Example 1 of a triangle, its three bisectors and the incenter

Let ABC de a triangle in which a=3 cm, b=4 cm and c=2 cm. Find the angle bisectors Ba, Bb y Bc.

SOLUTION:

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

Formula of the three angle bisectors of the triangle

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

Calculation of the semiperimeter in example 1

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

Solution in example 1

Therefore, Ba=2.45 cm, Bb=1.47 cm and Bc=3.32 cm.


AUTHOR: Bernat Requena Serra

YEAR: 2020


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