Question Video: Recognising Facts About Circumcircle Construction Mathematics • Third Year of Preparatory School

Video Transcript

True or False: The perpendicular bisectors of the sides of a triangle intersect at a point that is the center of the circumcircle of the triangle.

Perhaps the main obstacle to solving this question is understanding what the terminology means.

Let us start off by dissecting the phrase “perpendicular bisectors of the sides of a triangle.” To illustrate this, let us draw a random triangle. If we consider one of the sides, then the perpendicular bisector to that side is a line that bisects it, i.e., divides it in two, at a perpendicular angle. We can find this by first finding the midpoint of the side and then by drawing a line through that point that is at a right angle to the side.

Continuing onwards with the question, it is stated that the perpendicular bisectors intersect at a point and that this point is the center of the circumcircle of the triangle. Recall that a circumcircle of a triangle is a circle that passes through each of the vertices, also known as the corners, of the triangle.

So it is postulated that if we drew a perpendicular bisector through every side and found their intersection point, then this point would be the center of a circumcircle. However, even if we did this for our example, and we found that the circle was a proper circumcircle, this would not prove the statement for all cases. It would only prove it for one specific case. So, in order to prove the statement is true or false properly, we need to construct a mathematical argument. This can be done by using what we know about perpendicular bisectors and circles.

Recall that a circle is a set of points equidistant, which means at a constant distance, from the center. This distance is the radius, 𝑟. For the case of a circumcircle, this includes the three vertices of the triangle. They too must be equidistant from the center.

Next, we recollect that all points on the perpendicular bisector of a line segment 𝐴𝐵 are equidistant from the endpoints 𝐴 and 𝐵. This is significant because in this case the endpoints are the vertices of the triangle. So, if we were to label two of the vertices of our triangle 𝐴 and 𝐵, then all points lying on the perpendicular bisector, including the intersection point with the other bisectors, are equidistant from 𝐴 and 𝐵.

We can then repeat this logic for the other bisectors. If we label the third vertex 𝐶, then all points on the bisector of 𝐴𝐶 are equidistant from 𝐴 and 𝐶. And the same goes for the points 𝐵 and 𝐶.

Therefore, the intersection point is equidistant from all three vertices. This means by definition that this point is the center of the circumcircle. Therefore, the answer is true.