Can a secant-secant angle ever be obtuse?

Question

If two secants to a circle are drawn from the same exterior point, can the angle formed between the 2 secants ever be obtuse (or right)?

Thank you

Answer 1

Draw your favorite obtuse angle with apex point $D$. Draw its bisector and choose a point $A$ on the bisector.

Drop an altitude $AE$ from the bisector to one of the legs of the original triangle. Draw the circle with center $A$ that touches the two legs of the angle. Its radius $AE$ will be smaller than $AD$.

Now draw a second circle also centered on $A$ with a radius between $AE$ and $AD$. This circle has the legs of your obtuse angle as secant lines.

(If you wanted to start with a particular circle, you can now scale and translate the entire figure such that the last circle you drew coincides with the one you want).

Answer 2

Here's to give you an intuition:

Let there be a point $D$ outside a circle, such that $DC$ and $DE$ are tangents, with $A$ as center of the circle. Radius of the circle is $R$.

Note that $$\tan \theta = \frac{R}{\sqrt{S_1}}$$

The angle between the two tangents is $2\theta=2\tan^{-1}\frac{R}{\sqrt{S_1}}$.

Since $\sqrt{S_1}\in [0, \infty)$, hence, the angle between the two tangents can take acute, right, or obtuse values, depending upon the position of $D$ wrt the circle.

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Now, of what line form does the tangent happen to be a limiting case of?