CAT Circle through Three Points - Practice Questions & MCQ

Take a point A on a paper and draw a circle that can pass through this point A. You see that there can be as many circles that can pass through this point A. So, Infinite number of circles can be drawn passing through A.

Now take two points A and B. You again see that there may be an infinite number of circles passing through A and B.

Now take three points A, B and C which are not on the same line. Draw a circle passing through these points A, B and C. Only one circle can pass through these points.

Theorem 5 : There is one and only one circle passing through three given non-collinear points.

Take three points A, B and C, which are not on the same line. Jion AB and BC. Draw perpendicular bisectors of AB and BC say, PQ and RS respectively.

Let these perpendicular bisectors intersect at one point O. (Note that PQ and RS will intersect because they are not parallel)

Now O lies on the perpendicular bisector PQ of AB, we have,

                         OA = OB

Again, O lies on the perpendicular bisector RS of BC, we have

                         OB = OC

Thus,           OA = OB = OC,

Which means that the points A, B and C are at equal distances from the point O.

So if you draw a circle with centre O and radius OA, it will also pass through B and C. This shows that there is a circle passing through the three points A, B and C.

As we know that two lines (perpendicular bisectors) can intersect at only one point, so you can draw only one circle with radius OA.

Note:

If ABC is a triangle, then by Theorem 5, there is a unique circle passing through the three vertices A, B and C of the triangle. This circle is called the circumcircle of the ∆ ABC. Its centre and radius are called respectively the circumcentre and the circumradius of the triangle.