With the coordinates of the four points p1, p2, p3, and p4, where p1, p2, and p3 lie on a circle, find whether p4 lies inside or outside of the circle.

Question

Accepted Answer

To determine if p4 lies inside or outside the circle with p1, p2, and p3 coordinates.

Calculate the equation of the circle using p1, p2, and p3 coordinates.
Check the distance between p4 and the center of the circle.
If the distance is less than the radius of the circle, p4 lies inside the circle, otherwise outside.

Accepted Answer

The question is to check your understanding of analytical geometry and vector algebra.

The short answer to the question is as follows:

Need to find the circle center from the XYZ coordinates of the points p1, p2, and p3.
Find the distance between the center and point p1 (or points p2 or p3 that lie on the circle) which is the radius of the circle.
Find the distance between the center and the point p4.
If the distance is greater than the radius of the circle, then the point is inside the circle otherwise outside the circle.

In case they ask you to go into detail about finding the center of the circle with three points:

Find two vectors from the three points like v1 = p2-p1 & v2 = p3-p1
Find dot products of the vectors v11 = v1^T . v1, v22 = v2^T . v2 & v12 = v1^T . v2.
Find the two scalars with b = 1/( 2 * (v11 . v22 - v12^2)) as, k1 = b. v22 (v11 - v12) & k2 = b. v11 (v22 - v12).
Finally, the point equation is P0 = p1 + k1 . v1 + k2 . v2
Only go this detailed for finding the center if they insist on hearing that.

With the coordinates of the four points p1, p2, p3, and p4, where p1, p2, and p3 lie on a circle, find whether p4 lies inside or outside of the circle.

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