Thanks for the link.  Keeping the window square is easy and works.
However, I am having problems with my method:

I added a double to the class called aspect_ratio.

In the constructor, I calculate

   aspect_ratio = static_cast<double>(image->Width) / static_cast<double>(image->Height);

Then in the Resize event handler:

System::Void PlotDisplayWindow_Resize(System::Object *  sender, System::EventArgs *  e)
{
   Control* control = __try_cast<Control*>(sender);

   // Looking at this, I think I need Height instead of Width, but the
   // concept is what's important
   control->Size =  System::Drawing::Size(static_cast<int>(aspect_ratio * control->Size.Width), control->Size.Width);
}

When I do this and try to resize the form, I get a flickering of
rectangles of the appropriate aspect ratio as I drag the border of the
frame, but as soon as I let go of the border then the window snaps back
to its original size.