Given a circle whose centre is "o" with 2 secants; "abc" and "aef" (both of them start in point a, which is out of the circle).
The radius is called $r$. Also, the value if $AB \cdot AC$ is known, ab*ac= M.
find the length of the line that connects the centre and point "a" (line "ao") by using only "M" and "r".
If it helps, the answer is $\sqrt{m+r^2}$. I need the solution method.
p.s. I hope I was clear enough, I don't have a rich English vocabulary regarding to math.
On
Ok. after I published it, it seemed a lot simpler.
I just draw a tnagent from "A" and connected a radius to it. Then, the angle between them is 90 deg' and we can represent AO as squrt of (R^2 + the tangent).
Now represent the tangent by using the statement that a tangent in the power of two is equal to the outside of a secant times all the secant. Which we already know as "M".
Now replace and that's it!
Let $AP$ be a tangent of the circle, where $P$ is a touching point.
Thus, $$AP\perp OP$$ and $$AP^2=AB\cdot AC=m,$$ which gives $$AP=\sqrt{m}$$ and $$AO=\sqrt{OP^2+AP^2}=\sqrt{r^2+m}.$$