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, 𝐵, and
𝐶. 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.