@Dkunb
There might be some real valued solutions the plot command misses.

Plotting your assumptions reveals that they are "over constrained" (i.e. too many).

assumptions := 1/5 < x, x < 1, 0 < beta, beta < 1, 1/2 < x/(beta*x^2 + 1);
plots:-inequal({assumptions}, x = 0 .. 1, beta = -0.1 .. 1, axes = frame);

I have tried to work with refined assumptions

assumptions_a := 1/2 < x, x < 1, 0 < beta, beta < 1; 
assumptions_b := x < 1, 0 < beta, beta < (2*x - 1)/x^2;

without success.

Maybe someone knows a magic trick to show that roots number 3 an 4 are imaginary

[{x = (-sqrt(3)*5^(2/3)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(3/4)*sqrt(beta) + sqrt(3)*5^(5/6)*sqrt(beta)*sqrt((-5*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + (34*beta - 100)*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + (-5*beta^2 + 130*beta - 125)*5^(2/3))*sqrt(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625) - 450*(sqrt(beta) + (4*beta^(3/2))/25)*sqrt(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)*5^(2/3)*sqrt(3)) + 45*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))/(150*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))}, {x = (-sqrt(3)*5^(2/3)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(3/4)*sqrt(beta) - sqrt(3)*5^(5/6)*sqrt(beta)*sqrt((-5*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + (34*beta - 100)*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + (-5*beta^2 + 130*beta - 125)*5^(2/3))*sqrt(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625) - 450*(sqrt(beta) + (4*beta^(3/2))/25)*sqrt(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)*5^(2/3)*sqrt(3)) + 45*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))/(150*beta*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/6)*(5*5^(1/3)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(2/3) + 5^(2/3)*(17*beta - 50)*(5*beta^3 + 625 + 18*sqrt(beta)*sqrt(40*beta^4 + 3674*beta^3 + 22575*beta^2 + 29375*beta + 15625) + 1101*beta^2 + 3075*beta)^(1/3) + 25*beta^2 - 650*beta + 625)^(1/4))}]

  Even then only the first root fits to your assumptions (see the plot above).
 

More observations that were repeatable at least 3 times:

• A successfully loaded classic worksheet (*.mws) saved under to the *.mw format could always been opened by double-click with 2023.
• When running Maple 2021 (or other instances of Maple) in parallel while the computer was on battery, the classic worksheet could be opened by double-click. Without other instances running in parallel Maple 2023 froze at the fist double-click.
• Back in office running the computer plugged in did not show a dependency on other instances of Maple running in parallel. This time Maple 2023 froze on the second double-click after a first successful attempt to open by double-click (Maple was closed before the second double-click).  The second double-click to freeze Maple 2023 was a new behavior.
• After logging out and in, no freezes occurred. At this point, I saw myself in my mind being taken away in a straitjacket, screaming, but it really happened. I decided to discontinue my investigations and opened other applications. For some reason I double clicked one last time and Maple 2023 froze again. Finally, it turned out that Maple froze when a pdf reader (Foxit) was running in parallel.   
• I decided a reboot and (unplanned) Windows was updated. Since then, I can’t force a freeze anymore.

If this case were repeatable, I would ask differently now: Why does Maple 2023 freeze under certain states of the computer system when opening this particular classic worksheet

With the update of Windows on my computer, this case is now a cold case

@mmcdara 

I get squares on my computer with 2023. Can you send a screen shot how it should look like?

Update: It looks like this after opening

@all: Is that reproducible on other computers or is it only my local installation?

@Nicole Sharp 
I do not think it’s a bug its more an enhancement you are looking for which I would also welcome for text passages. If you want to do math with it, you could try in the meatime the following:

Download userdefined_symbols.mw

And yes, fun is important in life! 🙂

@mmcdara 
Kitonum is correct. I used jargon from CAD where intersection is the operation to drill holes into a body, for example.
Thank you for providing the intersection line that I needed as well. Both answers fit nicely together.
@Kitonum 

To express my reaction in HTML: &#x1f632
Can you give me hint what max does with implict geometries?

I am missing something important here.
min, by the way, leaves out the inner surfaces which is nice as well.

Nice, but not reproducible for me (with modified color option)

a := 5;
b := 1;
h := 7;
alpha := Pi/2 + 1/tan(3/2);
plot3d([(a + b*cos(u))*cos(v) - b*cos(alpha)*sin(v)*sin(v), a + b*cos(u)*sin(v) + b*cos(alpha)*sin(v)*cos(v), b*sin(alpha)*sin(u) + h*v], u = -2*Pi .. 2*Pi, v = -2*Pi .. 2*Pi, numpoints = 2000, style = surface, color = "DarkRed");

Could you share the plot command you used?

As vv and sand15 indicated in an earlier post it would be nice to get a little more background.

@Nicole Sharp 

I agree, renaming (and hiding labels) would clean up documents without the need of unnecessary temporary variables.

What might interest you: Hiding is hiden but possible.

@Thomas Richard 

It's a specific file. I will send it.

The freeze looks like that (with a Windows hourglas not visible on the screen shot):

The taks has to be killed.

Thank you

@Thomas Richard 

One reason it is less popular may be the description. I am one of those users for whom "evaluate in an algebraic number (or function) field" does not indicate that this command is useful for simplification tasks of expressions.
I will experiment with it.

@QM I could not resist to ask the new kid on the block

I skip the intermediate steps

I am not sure if this "simplification" is correct. At least I could not simplify it with Maple

;-)

@acer did mean that in your first code snippet?

cot(Pi/2 - alpha/2);
                             /1      \
                          tan|- alpha|
                             \2      /

@Thiago_Rangel7 
Convert comes with new options for trig functions. In this case

convert(1/cot(a), tan);
                             tan(a)

does the "simplification". There is also sincos and more.

@Kitonum 

Both ways are of value for me. I will use the way of @ecterrab for old code where I solved for sin and cos to get the arguments for arctan(y, x).

For new code, convert is better if the code is shared between platforms.

Thank you both

Can you say for which images? I could imagine that such a function would be useful for background images of plots and image outputs of imagetools routines.

@acer 

I see the complication. I have to go back to cases where I retrospectively realized that a warning would have helped. Maybe I can come up with something better than “always working” (which I definitely do not want).

What still surprises me is the dependence on the input mode and the way the restart is performed. I have made more observations on that.