Calculate the value of AN

Question

In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

My idea was to form the triangle $ AMN $ which by the way is rectangle. So $ MN ^ 2 = AM ^ 2 + AN ^ 2 $

$ x ^ 2 = MN ^ 2-AM ^ 2 $

Answer 1

Let $O'$ be the center of $\gamma'$, the circle symmetrical to $\gamma$ with respect to $AN$.

1. $\triangle O'BN$ is isosceles, with $\overline{O'N}=\overline{O'B} = 5$, and $\overline{BN} = 6$.
2. Pythagorean Theorem gives $\overline{O'K} = \overline{AH}=4$.
3. Euclid on $\triangle AMN$ yieds $$\begin{cases}\overline{MH}\cdot\overline{HN} = 16\\\overline{MH}+\overline{HN} = 10.\end{cases}$$Thus $\overline{HN} = 8$.
4. Pythagorean Theroem again gives $\overline{AN}^2 = 64+16=80$.