Question 2

We will use properties of angles subtended by a chord in a circle.

Step 1 — Analyze Part (i)

Let's consider points $X$ and $Y$ on the circle. They are on the same side of chord $AB$.

Angles subtended by the same chord in the same segment are equal. Chord $AB$ subtends $\angle AXB$ and $\angle AYB$. Since $X$ and $Y$ are on the same side of $AB$, they are on the same arc. Therefore, $\angle AXB$ must be equal to $\angle AYB$. There are no points $X, Y$ on the same side of $AB$ where these angles differ.

Step 2 — Analyze Part (ii)

Let's assume $\angle AXB = \angle AYB$. We need to check if $X$ and $Y$ must be on the same side of chord $AB$. Consider $X$ on the major arc $AB$. Consider $Y$ on the minor arc $AB$. If $X$ and $Y$ are on opposite sides of $AB$, then $AXBY$ forms a cyclic quadrilateral. In a cyclic quadrilateral, opposite angles are supplementary. So, $\angle AXB + \angle AYB = 180^\circ$. If $\angle AXB = \angle AYB$, then $2 \angle AXB = 180^\circ$. This means $\angle AXB = 90^\circ$. In this specific case, if $AB$ is a diameter, then $\angle AXB = 90^\circ$ and $\angle AYB = 90^\circ$. Here, $\angle AXB = \angle AYB$, but $X$ and $Y$ are on opposite sides of $AB$. So, $X$ and $Y$ do not always lie on the same side of $AB$.

Step 3 — Analyze Part (iii)

We are given that $\angle AXB = \angle AYB$. Also, $X$ and $Y$ do not lie on the original circle. We need to check if the circle through $A, B, X$ also passes through $Y$. This means we need to check if points $A, B, X, Y$ are concyclic. The converse theorem of angles in the same segment states: If a line segment ($AB$) subtends equal angles ($\angle AXB = \angle AYB$) at two points ($X$ and $Y$) on the same side of the line segment, then the four points ($A, B, X, Y$) are concyclic. Assuming $X$ and $Y$ are on the same side of $AB$, this theorem applies. Therefore, points $A, B, X, Y$ lie on the same circle. So, the circle passing through $A, B, X$ will also pass through $Y$.

Answer

(i) No. (ii) No, this is not always true. (iii) Yes.