11.3. Two circles intersect at points A and B. A tangent to the first circle is drawn through point A, intersecting the second circle at point C. A tangent to the second circle is drawn through point B, intersecting the first circle at point D. Find the angle between the lines $\mathrm{AD}$ and $\mathrm{BC}$.
Answer. The lines are parallel, the angle is $0^{\circ}$.

Solution. Mark point $\mathrm{P}$ on the extension of $\mathrm{CA}$ beyond point A and point M on the extension of DB beyond point B. The inscribed angle ABD in the first circle, which subtends the chord AD, is equal to the angle PAD between the chord AD and the tangent AC to the first circle. The inscribed angle CAB in the second circle, which subtends the chord BC, is equal to the angle CBM between the chord BC and the tangent BD to the second circle. Then, the angle BAD between the secant line AB and the line AD is equal to the difference of $180^{\circ}$ and the sum of angles PAD and CAB, which is equal to the difference of $180^{\circ}$ and the sum of angles ABD and CBM, that is, the angle ABC between the secant line AB and the line BC. Therefore, the lines AD and BC are parallel.

![](https://cdn.mathpix.com/cropped/2024_05_06_d12621c3941bc2007d1fg-1.jpg?height=511&width=619&top_left_y=1818&top_left_x=1301)

Grading criteria. Only answer: 0 points.