Construct a circle of a given radius passing through a given point and tangent to a given line.

#
The center of the desired circle is at a given distance from a given line and from a given point.

## Solution

The center of the desired circle is at the intersection of a line parallel to the given one and at a distance equal to the given radius, and a circle with the given radius, whose center is the given point.

![](https://cdn.mathpix.com/cropped/2024_05_06_4fce0705b02fa9e106d4g-02.jpg?height=106&width=1424&top_left_y=113&top_left_x=14)

$A B C$ is a secant, $A$ is an external point of the circle, the angular measure of arc $B D$ is $42^{\circ}$, and the angular measure of arc $B D C$ is $220^{\circ}$. Find the angle $A B D$.

## Hint

An inscribed angle is equal to half the angular measure of the arc on which it stands.

## Solution

Angle $A B D$ is adjacent to angle $D B C$, which is measured by half the arc $D C$ not containing point $B$, i.e.,

$$
\angle D B C=\frac{1}{2}\left(220^{\circ}-42^{\circ}\right)=89^{\circ}
$$

Therefore, $\angle A B D=91^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_4fce0705b02fa9e106d4g-02.jpg?height=454&width=463&top_left_y=842&top_left_x=800)

## Answer

$91^{\circ}$.