Video Transcript
In the following two figures, two
types of construction have been made on the same triangle 𝐴𝐵𝐶. Which point will be the center of
the circle that passes through the triangle’s vertices?
In this question, we’re asked to
determine the center of a circle which passes through all three of the triangle’s
vertices. That’s the points 𝐴, 𝐵, and
𝐶. And to do this, we need to
remember, the center of a circle will be equidistant from all points on the
circumference of the circle. Therefore, we need to determine
which points in either of the two diagrams is equidistant from all three points 𝐴,
𝐵, and 𝐶. To do this, we know in the second
diagram we’re given the perpendicular bisectors of all three sides of the
triangle. We can see that these are the
perpendicular bisectors because they cut the sides of the triangle at right
angles. And they cut the sides of the
triangle in half represented by either the one, two, or three lines.
This is useful because every single
point on the perpendicular bisector between 𝐵 and 𝐶 will be equidistant from 𝐵
and 𝐶. For example, the point 𝐹 is
equidistant from both 𝐵 and 𝐶. This result is true for any
perpendicular bisector, so we can see that 𝐹 also lies on the perpendicular
bisector between 𝐴 and 𝐵. So 𝐹 is also equidistant from 𝐴
and 𝐵. This means it’s equidistant from
all three vertices of the triangle. Therefore, if we draw a circle
centered at 𝐹 of radius the distance between 𝐴 and 𝐹, we would get a circle which
passes through all three vertices of the triangle.
The same cannot be true for the
point 𝑁; although it’s found by taking the bisectors of each of the angles of our
triangle, 𝑁 will not necessarily be equidistant from all three vertices of the
triangle. Therefore, we can conclude the
point 𝐹 is the center of a circle which passes through all three vertices of the
triangle.
