Question

Consider four points on a circle. The length of the line segment form by the two adjacent points is less than the diameter of the circle, and it subtends equal angle at the other two points. Determine the relative position of the $3^{rd}$ and $4^{th}$ points.

• A. They are on the same side of the line segment.
• B. They can be on either side of the line segment.

Answer 1

The correct option is A They are on the same side of the line segment.

We know that if the line segment joining two points subtends equal angles at two other points lying on the same side of the line, then all the four points lie on a circle.

Let the first two points be A and B, and the $3^{rd}$ and $4^{th}$ points are C and D, respectively.

As A and B form a line segment of less than the circle's diameter, C and D must lie on the same side of the line.

If we place points A and B at end points of diamters of the circle, then $\angle C$ and $\angle D$ will be at right angles and can be present on either side of the line segment.