Subfield: Geometry


Question: In the diagram, two circles are tangent to each other at point $B$. A straight line is drawn through $B$ cutting the two circles at $A$ and $C$, as shown. Tangent lines are drawn to the circles at $A$ and $C$. Prove that these two tangent lines are parallel.

![](https://cdn.mathpix.com/cropped/2023_12_21_7e10dbc5f8ae70c21b8eg-1.jpg?height=366&width=531&top_left_y=657&top_left_x=1252)

Solution:
Let the centres of the two circles be $O_{1}$ and $O_{2}$.

Join $A$ and $B$ to $O_{1}$ and $B$ and $C$ to $O_{2}$.

Designate two points $W$ and $X$ on either side of $A$ on one tangent line, and two points $Y$ and $Z$ on either side of $C$ on the other tangent line.

![](https://cdn.mathpix.com/cropped/2023_12_21_87da1d35d2fbe47e109eg-1.jpg?height=375&width=618&top_left_y=1474&top_left_x=859)

Let $\angle X A B=\theta$.

Since $W X$ is tangent to the circle with centre $O_{1}$ at $A$, then $O_{1} A$ is perpendicular to $W X$, so $\angle O_{1} A B=90^{\circ}-\theta$.

Since $O_{1} A=O_{1} B$ because both are radii, then $\triangle A O_{1} B$ is isosceles, so $\angle O_{1} B A=$ $\angle O_{1} A B=90^{\circ}-\theta$.

Since the two circles are tangent at $B$, then the line segment joining $O_{1}$ and $O_{2}$ passes through $B$, ie. $O_{1} B O_{2}$ is a straight line segment.

Thus, $\angle O_{2} B C=\angle O_{1} B A=90^{\circ}-\theta$, by opposite angles.

Since $O_{2} B=O_{2} C$, then similarly to above, $\angle O_{2} C B=\angle O_{2} B C=90^{\circ}-\theta$.

Since $Y Z$ is tangent to the circle with centre $O_{2}$ at $C$, then $O_{2} C$ is perpendicular to $Y Z$. Thus, $\angle Y C B=90^{\circ}-\angle O_{2} C B=\theta$.

Since $\angle X A B=\angle Y C B$, then $W X$ is parallel to $Y Z$, by alternate angles, as required.
Let the centres of the two circles be $O_{1}$ and $O_{2}$.

Join $A$ and $B$ to $O_{1}$ and $B$ and $C$ to $O_{2}$.

Since $A O_{1}$ and $B O_{1}$ are radii of the same circle, $A O_{1}=B O_{1}$ so $\triangle A O_{1} B$ is isosceles, so $\angle O_{1} A B=\angle O_{1} B A$.

![](https://cdn.mathpix.com/cropped/2023_12_21_60a3f40dbae5bbe7d26dg-1.jpg?height=371&width=548&top_left_y=430&top_left_x=889)

Since $\mathrm{BO}_{2}$ and $\mathrm{CO}_{2}$ are radii of the same circle, $B O_{2}=C_{2}$ so $\triangle B O_{2} C$ is isosceles, so $\angle O_{2} B C=\angle O_{2} C B$.

Since the two circles are tangent at $B$, then $O_{1} B O_{2}$ is a line segment (ie. the line segment joining $O_{1}$ and $O_{2}$ passes through the point of tangency of the two circles).

Since $O_{1} B O_{2}$ is straight, then $\angle O_{1} B A=\angle O_{2} B C$, by opposite angles.

Thus, $\angle O_{1} A B=\angle O_{1} B A=\angle O_{2} B C=\angle O_{2} C B$.

This tells us that $\triangle A O_{1} B$ is similar to $\triangle B O_{2} C$, so $\angle A O_{1} B=\angle B O_{2} C$ or $\angle A O_{1} O_{2}=$ $\angle C O_{2} O_{1}$.

Therefore, $A O_{1}$ is parallel to $C_{2}$, by alternate angles.

But $A$ and $C$ are points of tangency, $A O_{1}$ is perpendicular to the tangent line at $A$ and $\mathrm{CO}_{2}$ is perpendicular to the tangent line at $C$.

Since $A O_{1}$ and $C O_{2}$ are parallel, then the two tangent lines must be parallel.

Final answer: 证明题，略