### 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 βperpendicular bisectors of the sides of a triangle.β 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,
π. 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 π΄π΅ are equidistant from the
endpoints π΄ and π΅. 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 π΄ and π΅, then all points lying on the perpendicular
bisector, including the intersection point with the other bisectors, are equidistant
from π΄ and π΅.

We can then repeat this logic for
the other bisectors. If we label the third vertex πΆ,
then all points on the bisector of π΄πΆ are equidistant from π΄ and πΆ. And the same goes for the points π΅
and πΆ.

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.