Finding Tangents

OK, this is really shameful, but I cannot for the life of me figure out something that ought to be rather simple.  Take a circle, X^2+Y^2=r^2, centered at the origin, and a point, (p,q), outside the circle.  There are two tangent lines to that circle through that point.  Find them.

I know that the Pythagorean theorem says that (p^2+q^2)=r^2+m^2, where m is the distance of (p,q) to either tangent point.  And certainly given any point on the perimeter of the circle, it is easy to determine if the tangent line through that point passes through (p,q).  But going the other way ... what am I missing?

Thank you!

can we use calculus

Whatever works, but do we need calculus to do this?

so you want the equations of the tangent lines?

Well, you could take that Pythagorean theorem, with the distance from the point to the circle center and the radius of the circle, this lets you get the distance between the point and where the tangent line hits the circle.

With this, you can take that distance, make it a radius, and let the point be the center and you get a second circle.

Find where these two circles (this one and the original one) intersect and thats your tangent point.

I haven't tried it, but a system of 2 equations, 2 unknowns should be relatively simple to solve.

There's probably an easier geometric solution than the algebraic one, but I wouldn't know it.

I have not actually taken the time to do it, but I suggest calling the 2 points on the circle (x1,y1) and (x2,y2), subscripts of course.  That should get you 2 equations in  3 unknowns right there, using the 2 point formula for a line and the values for the points on the circle from the circle equation.  Then, if you define the circle as 2 separate functions, we know the first derivative at those 2 points on the circle are the slopes of the tangent lines.  That might lead somewhere.  If not, try something else. LOL.

Thank you all, I shall try it!