MHB Can three tangents to a circle meet at a common point?

[poissonspot]

I considered the question of whether three tangents to the circle could meet at a common point and only came up with a contradiction by lengthy constructive means.
Circles are "nice", so there must be some clever ways of showing this fact that given a point outside the circle there are two tangent lines to the circle passing through this point. Any ideas? Thanks mucho,

 

[MarkFL]

You could, without loss of generality, use a unit circle, and a point on the $y$-axis outside of the circle $(0,b)$ where $1<b$.

The equation of the circle is:

$$x^2+y^2=1$$

The equation of the tangent lines is:

$$y=mx+b$$

Substitute for $y$ from the line into the circle:

$$x^2+(mx+b)^2=1$$

$$x^2+m^2x^2+2bmx+b^2=1$$

$$\left(m^2+1 \right)x^2+(2bm)x+\left(b^2-1 \right)=0$$

Now, by analyzing the discriminant, we find:

$$\Delta=(2bm)^2-4\left(m^2+1 \right)\left(b^2-1 \right)$$

$$\Delta=4\left(b^2m^2-b^2m^2+m^2-b^2+1 \right)$$

$$\Delta=4\left(m^2-b^2+1 \right)$$

We see that with $1<b$, there will be two real values of $m$, given by:

$$m=\pm\sqrt{b^2-1}$$

This implies there are two tangent lines:

$$y=\pm\sqrt{b^2-1}x+b$$

 

[poissonspot]

You could, without loss of generality, use a unit circle, and a point on the $y$-axis outside of the circle $(0,b)$ where $1<b$.

The equation of the circle is:

$$x^2+y^2=1$$

The equation of the tangent lines is:

$$y=mx+b$$

Substitute for $y$ from the line into the circle:

$$x^2+(mx+b)^2=1$$

$$x^2+m^2x^2+2bmx+b^2=1$$

$$\left(m^2+1 \right)x^2+(2bm)x+\left(b^2-1 \right)=0$$

Now, by analyzing the discriminant, we find:

$$\Delta=(2bm)^2-4\left(m^2+1 \right)\left(b^2-1 \right)$$

$$\Delta=4\left(b^2m^2-b^2m^2+m^2-b^2+1 \right)$$

$$\Delta=4\left(m^2-b^2+1 \right)$$

We see that with $1<b$, there will be two real values of $m$, given by:

$$m=\pm\sqrt{b^2-1}$$

This implies there are two tangent lines:

$$y=\pm\sqrt{b^2-1}x+b$$

Thanks, this is close to how I approached the problem. I used vector algebra though instead and solved for points where the dot product was equal to zero.

btw, I'm still interested in other solutions if anyone else is. Thx again,