Circumcenter of a triangle

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Drawing of the circumcenter of a triangle as the intersection of the three bisectors

The circumcenter of a triangle (O) is the point where the three perpendicular bisectors (Ma, Mb y Mc) of the sides of the triangle intersect. It can be also defined as one of a triangle’s points of concurrency.

The perpendicular bisector of a triangle is a line perpendicular to the side that passes through its midpoint.

Drawing the circumcenter as the center of the circumscribed circumference of a triangle

The circumcenter (O) is the central point that forms the origin of the circumcircle (circumscribed circle) in which all three vertices of the triangle lie on the circle.

It’s possible to find the radius (R) of the circumcircle if we know the three sides and the semiperimeter of the triangle.

The radius of the circumcircle is also called the triangle’s circumradius.

The formula for the circumradius is:

Formula for the radius of the circumscribed circle in the triangle with center at the circumcenter

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.

Where is the Circumcenter of a Triangle Located?

  • If it’s an obtuse triangle the circumcenter is located outside the triangle (as we see in the picture above).
    Drawing of the outer circumcenter to the triangle and the circumscribed circumference
  • If it’s an acute triangle the circumcenter is located inside the triangle.
  • If it’s a right triangle the circumcenter lies on the midpoint of the hypotenuse (the longest side of the triangle, that is opposite to the right angle (90°). We can see an example in the figure below.
    Drawing of the circumcenter in a right triangle

See the Thales’ Theorem.

Euler Line

In any non-equilateral triangle the orthocenter (H), the centroid (G) and the circumcenter (O) are aligned. The line that contains these three points is called the Euler Line.

Drawing of Euler's line

In an equilateral triangle all three centers are in the same place.

The relative distances between the triangle centers remain constant.

Distances between centers:

It is true that the distance from the orthocenter (H) to the centroid (G) is twice that of the centroid (G) to the circumcenter (O). Or put another way, the HG segment is twice the GO segment:

Formula for the relation of the distances between centers on the Euler line

When the triangle is equilateral, the centroid, orthocenter, circumcenter, and incenter coincide in the same interior point, which is at the same distance from the three vertices.

This distance to the three vertices of an equilateral triangle is equal to Distance 1 on Euler's Line from one side and, therefore, Distance 1 on Euler's Line to the vertex, being h its altitude (or height).

Exercise 1

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 2

Find the coordinates of the circumcenter of a triangle O ABC whose vertices are A(3, 5), B(4, -1) y C(-4, 1).

Drawing of the exercise 2 of the circumcenter of a triangle

SOLUTION:

In order to find the circumcenter O we have to solve the equations for two perpendicular bisectors Ma (perpendicular to side a) and Mb (perpendicular to side b) and see where is located the intersection point (that is the circumcenter O) of both perpendicular bisectors.

Drawing 2 of the exercise 2 of the circumcenter of a triangle

STEP 1: Find the equation for the perpendicular bisector Ma.

Firstly we will find the equation of the line that passes through side a, which is the opposite of vertex A. This equation is obtained knowing that it passes through points B (4, -1) and C (-4, 1). The general equation of the line that passes through two known points is:

Calculation of the line through two points in exercise 2

The equation of the line that contains side BC and its slope m will be:

Calculation of the line containing side BC in exercise 2

So, the slope m for the line a is -1/4.

Now, we get the coordinates of the midpoint r between vertices B and C, i.e. midpoint of side a.

Where is r?

Calculation of where is r in exercise 2

So,

Calculation of the midpoint r in exercise 2

The slope of the line that contains the perpendicular bisector Ma, being perpendicular to the side a, is the inverse and of the opposite sign to the slope of the line found that contains side a. So, we have that:

Calculation of the slope of the perpendicular in exercise 2

So, the slope of the line Ma is 4 because the slope of the line a it was -1/4.

Since we know that perpendicular bisector Ma passes through the midpoint r (located at (0, 0)) and we know its slope mp, which is equal to 4, now we can obtain the equation for the line Ma:

Calculation of the equation of the bisector Ma in exercise 2

This is the equation for the perpendicular bisector Ma.

STEP 2: Find the equation for the perpendicular bisector Mb.

Now we proceed in the same way to find the equation of the line that contains the perpendicular bisector Mb, that is, the one that passes through the midpoint s and is perpendicular to the side b between vertices A and C.

First, we calculate the slope of the line b (or side b):

Calculation of the slope of the line AC in exercise 2

Then we find the midpoint s coordinates between vertices A and C:

Calculation of the midpoint s in exercise 2

The equation of the line that contains the perpendicular bisector Mb, that is, the one starting from the midpoint s is perpendicular to side b. Therefore, the slope of this line will therefore be –7/4 (inverse and of the opposite sign). With the slope of a line and one of its points we can find the equation:

Calculation of the equation of the bisector Mb in exercise 2

We have the equations of two of the perpendicular bisectors of the triangle, Ma and Mb:

Drawing of the circumcenter and equations in exercise 2

Next, we solve this system of two equations in two variables using the substitution method, the most suitable, given the form of the first equation:

Calculation of the solution of equations in exercise 2

Finally, we have that x = 0,37 and y = 1,48.

Hence, circumcenter O (0.37, 1.48).

Drawing of the circumcenter solution in exercise 2

The circumcenter O is the centerpoint of the circumscribed circle:

Drawing of the circumscribed circumference in exercise 2

AUTHOR: Bernat Requena Serra

YEAR: 2020


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