420. A tangent to a circle was rotated by an angle $\beta$ around the point of tangency, causing it to become a secant. Find the measure of the arc it intercepts.
$\triangleright$ Let $OA$ be the radius drawn to the point of tangency, and $AB$ be the secant. The tangent is perpendicular to the radius, so the angle $OAB$ between the radius and the secant is $90^{\circ}-\beta$. The triangle $OAB$ is isosceles, so the second angle in it is also $90^{\circ}-\beta$, and the third angle (with the vertex at the center of the circle) is $2 \beta$. Thus, the tangent turned by an angle $\beta$ cuts off an arc of size

![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-126.jpg?height=369&width=397&top_left_y=358&top_left_x=1321)
$2 \beta$. $\triangleleft$

The statement of the previous problem can be formulated as follows: the angle between the tangent and the chord passing through the point of tangency is equal to half the arc cut off by the chord.

![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-126.jpg?height=377&width=348&top_left_y=754&top_left_x=1365)

![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-127.jpg?height=248&width=263&top_left_y=670&top_left_x=385)