Cover Four Circles

Geometry Level 3

There are 4 points on a plane such that, given any 3 of them, it is possible to draw a unit circle that contains all 3 of them (inside or on its circumference).

Is it also possible to draw a unit circle that contains all 4 of the points?

No, not necessarily Yes, always

Agnishom Chattopadhyay by Agnishom Chattopadhyay , 22, India

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2 solutions

Nashita Rahman
Feb 6, 2018

It is given that it is possible to draw a unit circle from any of the three points .

Let the four points be A,B,C and D . If we take any three points randomly say A,B and D , they will lie on the circumference of a unit circle . Similarly B,C,D will also lie on the circumference of a unit circle and so on . Thus all the four points have to lie on the circumference of the same unit circle . This implies that the unit circle will contain all the four points .

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Michael Mendrin
Apr 29, 2018

Given that each point must belong in an area common with at least 3 circles, as in the following diagram:

then it is seen that an unit circle (blue) will always be able to cover those common areas. This is not a complete proof. We can start the blue circle with one of the black circles, and then move it outward until it covers the one other common area, without failing to fully cover the other 3.

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