Question Video: Understanding How to Construct a Circle Through Three Points Mathematics

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?

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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.

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