Question

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

• A. $\text{Circle}$
• B. $\text{Square}$
• C. $\text{Triangle}$
• D. $\text{Hexagon}$

Answer 1

The correct option is A $\text{Circle}$

There are $4$ points $\text{A}$, $\text{B}$, $\text{C}$ and $\text{D}$. Line segment $\text{CD}$ subtends at $\angle \text{DAC}$ and $\angle \text{DBC}$ on the points $\text{A}$ and $\text{B}$ respectively.
According to the question statement,
$\angle \text{DAC}=\angle \text{DBC}$

Note: The angles at the circumference subtended by the same arc are equal.
Here, $\text{CD}$ is the arc with chord as $\text{CD}$ and $\angle \text{DAC}=\angle \text{DBC}$.
So, the all $4$ point will lie on the circle.