Tangent intersection vs. radius?

[lifesmaelstrom]

Let's say two tangents to one circle intersect outside the circle at a point I'll call A. If the angle at which the two tangents intersect is under 90 degrees, that must mean that the intercepted arc is over 90; the radius is not 1. Drawing a line from A to the midpoint would intersect the closest point on the circle to A.

What is the relationship between the angle at A, the distance from A to the nearest point on the circle, and the radius? There has to be one..

Thanks.

 

[TchrWill]

Let's say two tangents to one circle intersect outside the circle at a point I'll call A. If the angle at which the two tangents intersect is under 90 degrees, that must mean that the intercepted arc is over 90; the radius is not 1. Drawing a line from A to the midpoint would intersect the closest point on the circle to A.

What is the relationship between the angle at A, the distance from A to the nearest point on the circle, and the radius? There has to be one..

Draw yourself a picture of a circle of radius R and center O, 2 tangents at points B and B' on the circle, and intersecting each other outside the circle at A and an angle of µ.
Draw line AO intersecting the circle at point C.
C is the point on the circle closest to point A.

Then, AO = R/sin(µ/2)and AC = R/sin(µ/2) - R and
AC = R[1/sin(µ/2) - 1].

Angle AOB = angle AOB' = 90 - µ/2.

If angle BAB' = 90º, then angle BOB' = 90º and arc BCB' is 1/4 of the circumference.

 

[Deleted member 4993]

Let's say two tangents to one circle intersect outside the circle at a point I'll call A. If the angle at which the two tangents intersect is under 90 degrees, that must mean that the intercepted arc is over 90; the radius is not 1. Drawing a line from A to the midpoint (midpoint of what?) would intersect the closest point on the circle to A.

What is the relationship between the angle at A, the distance from A to the nearest point on the circle, and the radius? There has to be one..

Thanks.