8. $A, B, G$ are three points on a circle with $\angle A G B=48^{\circ}$. The chord $A B$ is trisected by points $C$ and $D$ with $C$ closer to $A$. The minor arc $A B$ is trisected by points $E$ and $F$ with $E$ closer to $A$. The lines $E C$ and $F D$ meet at $H$. If $\angle A H B=x^{\circ}$, find $x$.
(1 mark) $A 、 B 、 G$ 是一個圓上的三點, 其中 $\angle A G B=48^{\circ}$ 。弦 $A B$ 被 $C 、 D$ 兩點三等分, 其中 $C$ 點離 $A$ 較近。劣弧 $A B$ 被 $E 、 F$ 兩點三等分, 其中 $E$ 點離 $A$ 較近。直線 $E C$ 和 $F D$ 相交於 $H$ 。若 $\angle A H B=x^{\circ}$, 求 $x$ 。
8. 32
8. Note that the point $G$ is not useful except that it tells us that the chord $A B$ subtends an angle of $48^{\circ} \times 2=96^{\circ}$ at the centre of the circle (which we call $O$ ) so that $\angle E O F=96^{\circ} \div 3=32^{\circ}$. We claim that $H A / / O E$ and $H B / / O F$. Assuming the claim, then $\angle A H B=\angle E O F=32^{\circ}$ and hence the answer would be 32 .
It remains to prove the claim. To show that $H A$ $/ / O E$, we shall prove that $A F$ is perpendicular to both $H A$ and $O E$. That $A F \perp O E$ is clear, because $E$ is the mid-point of arc $A F$. Now produce $H A$ and $F E$ to meet at $K$. First note that $A E=E F$, and $A C=C D$ implies $K E=E F$. In other words, the circle with diameter $K F$ passes through $A$, so $A F \perp H A$. This shows $H A / / O E$,
and by the same argument we have $H B$ // OF, thereby establishing the claim.