## 14. "Revolutionary" ${ }^{1}$ Geometry

Let in triangle $A B C$ angle $B$ be a right angle, and $M$ be a point on the hypotenuse, equidistant from the two sides of the triangle. Could you find the value of the following expression:

$$
\begin{aligned}
E & =\sqrt{1830}\left(A C-\sqrt{A B^{2}+B C^{2}}\right) \\
& +1789-\frac{\frac{1}{A B}+\frac{1}{B C}-\frac{\sqrt{2}}{B M}}{(1848)^{3}} ?
\end{aligned}
$$
14. 15) By the Pythagorean theorem $A C^{2}=A B^{2}+B C^{2}$; therefore,

$$
A C-\sqrt{\overline{A B^{2}+B C^{2}}}=0
$$

2) The area of $A B C=$ area of $A B M+$ area of $B M C$. Therefore, if point $M$ is at a distance $d$ from the legs of the triangle, then

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-157.jpg?height=383&width=805&top_left_y=1296&top_left_x=617)

$A B \cdot B C / 2=A B \cdot d / 2+B C \cdot d / 2$,

or

$$
1 / d=1 / B C+1 / A B
$$

But $B M=d \sqrt{2}$; therefore,

$$
1 / A B+1 / B C-\sqrt{2} / B M=0
$$

3) From the two previous observations, it follows that

$$
E=1789 \text {. }
$$