Question Video: Recognising Facts About Circle Construction Mathematics

True or False: If a circle passes through three points, then the three points should belong to the same line.

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Video Transcript

True or false: if a circle passes through three points, then the three points should belong to the same line.

To answer this question, let’s start by sketching a circle. We can choose three points on this circle. Let’s label these 𝐴, 𝐡, and 𝐢. We can see that these three points do not lie on the same straight line. However, the circle does pass through these three points. This is enough to prove the statement is false. If a circle passes through three points, then the three points do not need to belong to the same line.

This could make us ask an interesting question. What would happen if the three points were in a straight line? For example, can we construct a circle between these three points 𝑃, 𝑄, and 𝑅? The center of the circle must be equidistant from both three of these points. In particular, the center of the circle must be equidistant from both 𝑃 and 𝑄. And we know that the perpendicular bisector of 𝑃 and 𝑄 contains all points equidistant from both 𝑃 and 𝑄. So the center of our circle must lie on this line. Similarly, the center of this circle is equidistant from 𝑄 and 𝑅, so it must lie on the perpendicular bisector of 𝑄 and 𝑅.

And then we see a problem. These two lines are parallel. They never intersect, so there’s no point equidistant from 𝑃, 𝑄, and 𝑅. This leads to a slightly stronger result. There is no circle which passes through three points which lie on the same straight line. But we can answer this question as false. A circle that passes through three points do not need to have the three points lie on the same straight line. The statement is false.

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