Video Transcript
True or false: two distinct circles
can intersect at more than two points.
To answer this question, let’s
start by thinking about how circles can intersect. There’s two different ways that
circles can’t intersect. For example, we could have one
circle inside another circle, or we could just have two circles next to each
other. It’s also possible for two circles
to intersect at a single point, and in fact, there’s two different ways this can
happen. We can have both circles next to
each other, or we can have one circle inside of another circle. Both of these only intersect at a
single point. Finally, it’s also possible for
circles to intersect at two distinct points, for example, the following shape which
looks like a Venn diagram.
But what if we wanted two circles
which intersect at three distinct points? Well, if the three points of
intersection were noncolinear, then we know there’s a unique circle which passes
through all three of these points. This is called the circumcircle of
triangle 𝐴𝐵𝐶, and since it’s unique, we can’t have two distinct circles in this
case. But we’re not done yet. We also need to consider the case
if the three points of intersection lied on the same straight line. But this is also not possible
because we recall there is no circle which passes through three points which lie on
the same straight line. Therefore, we’ve shown the answer
is false. Two distinct circles cannot
intersect at more than two points.
