# equation for the radius of a circle that is tangent to two lines and passing through a specific point on one of the lines?

I'm interested in finding the equation for the radius (and optionally the center point) for a circle that is tangent to two lines and passing through a specific point on one of the lines. So far, I've found this tutorial showing the circle tangent to 2 lines and another circle.

http://www.arcenciel.co.uk/geometry/Circle2LC.pdf

I tried putting all those formulas in excel and making the circle have a radius of 0 for the point on a line, but it wasn't working as expected. Any help would be appreciated. Thanks!

Let $L$ be the first line and $(p,q) \in L$ the point to which you wish your circle to be tangent. Write down a parameterization of the line $L^\perp$ that is perpendicular to $L$ and passes through $p$: if $L$ is the solution of $ax+by=c$ then $L^\perp$ is parameterized as $$\gamma(t) = (p+at,q+bt)$$ Note that the distance from $\gamma(t)$ to $p \in L$ is $t \sqrt{a^2+b^2}$.
Starting from an equation for the second line $M$, in linear algebra you learn a formula for the function $P(x,y)$ which gives the orthogonal projection of a point $(x,y)$ to the line $M$. Form the equation $P(\gamma(t)) = t \sqrt{a^2+b^2}$, solve it for $t$, and you're done: $\gamma(t)$ is the center; and $t \sqrt{a^2+b^2}$ is the radius.