Four Points lying on the Circle

Q. 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 (will/ may not) lie on a circle.

Q. Consider the following statement:

If a line segment joining two points subtends equal angles at two other points lying on the either side of the line, then all the four points will lie on a circle.

The above statement is valid for which of the following cases?

1. Length of the said line segment is equal to radius of the circle.
2. Length of the said line segment is half the radius of the circle.
3. Length of the said line segment is twice the radius of the circle.
4. The statement is incorrect.

Q. Four poles are to be erected in a garden such that two poles form a line segment and the other two poles, lying on the same side of the line segment, are subtended at equal angles from the line segment. With the mentioned condition, the four poles can lie on a -

1. square
2. rectangle
3. circle

Q. Is the figure a circle-inscribed quadrilateral?

1. False
2. True

Q. 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.

1. They are on the same side of the line segment.
2. They can be on either side of the line segment.

Q. A line segment $UV$ of length $6\text{units}$ subtend equal angles at two points A and B lying on the same side of the line segment. A perpendicular of length $4\text{units}$ is dropped on the line segment from a fixed point whose distances from points $A,B,U\&V$ are same. Find the sum of distance of the points A and B from the fixed point.

1. $10\text{units}$
2. $5\text{units}$

Q. If two points on a line segment subtend identical angles at two other points lying on the same side of the line, then all four points will lie on

1. $\text{Circle}$
2. $\text{Square}$
3. $\text{Triangle}$
4. $\text{Hexagon}$