Constructing Modified Euler

Using the same strategy as done in the construction of Modified Euler. Starting from Trapezoidal Method $$y_1 = y_0 + \dfrac{h}{2}\left(f(x_0,y_0) + f(x_1,y_1)\right)$$ then approximating $f(x_1,y_1)$ using the explicit euler $y_1 = y_0 + hf(x_0,y_0)$ , thereby constructing the modified euler method $$y_1 = y_0 + \dfrac{h}{2}\left(f(x_0,y_0) + f(x_1,y_0 + hf(x_0,y_0))\right)$$

This is a valid method that is often seen in most textbooks.

New Method: Combining both backward and forward Euler methods

However, I wanted to ask why haven't I seen a similar approach done for the backward euler?

starting with the implicit euler:
$$y_1 = y_0 + hf(x_1,y_1)$$

then using the same approach as before, we can replace $y_1$ $$y_1 = y_0 + hf(x_1,y_0 + hf(x_0,y_0))$$

Come to think of it, maybe it is not used because it defeats the purpose of the implicit euler method. But I am not sure if this really the case. And, Is this even consistent?

I was thinking that it may be equivalent to some other method, and I should look for its equivalent RK-Method/Butcher Tableau?