在图中，P、Q、R 和 S 是圆上的点。PR 和 QS 相交于 T。

它不会直接告诉你答案，而是问你卡在哪一步看看 AI 是怎么陪你一步步弄懂这道题的

完整解法

1. Identify one pair of equal angles at the intersection of the chords.

Since $PR$ and $QS$ intersect at $T$ ,

$\angle PTS = \angle QTR$

because they are vertically opposite angles.

2. Identify a second pair of equal angles using the circle theorem.

Since $T$ lies on $QS$ , $\angle PST = \angle PSQ$ .

Since $T$ lies on $PR$ , $\angle QRT = \angle QRP$ .

Both $\angle PSQ$ and $\angle QRP$ are angles subtended by the same chord $PQ$ at the circumference.

Therefore,

$\angle PST = \angle QRT$

because angles in the same segment are equal.

3. Conclude the triangles are similar by AA similarity.

Since two pairs of corresponding angles are equal,

$\triangle PTS \sim \triangle QTR$

by AA similarity.

答案

$\angle PTS = \angle QTR \quad \text{(vertically opposite angles)}$

$\angle PST = \angle QRT \quad \text{(angles in the same segment, subtended by chord } PQ\text{)}$

Hence, since two pairs of corresponding angles are equal,

$\triangle PTS \sim \triangle QTR \quad \text{(AA similarity).}$

1. Use the similarity from part (a) to form a ratio of corresponding sides.

From part (a), $\triangle PTS \sim \triangle QTR$ .

Corresponding sides are proportional, so

$\frac{PT}{QT}=\frac{ST}{RT}.$

2. Cross-multiply to obtain the required product equality.

Cross-multiplying,

$PT \times RT = QT \times ST.$

Therefore,

$QT \times ST = PT \times RT.$

答案

Since $\triangle PTS \sim \triangle QTR$ ,

$\frac{PT}{QT}=\frac{ST}{RT}$

Hence,

$QT \times ST = PT \times RT.$

AI 分析，可能出错

用 AI 批改，圈出写错的那一步，并自动将你的薄弱点存入错题本，形成你的错因库。