Using a category named "PROJECTS" as menu button and post page.
Inserted image in a post and checked the PROJECTS category box.
Picture appears, as expected, in preview. . . but when published to site, text appears in post but no picture.

Sound like the project button takes you to a category multi post page and you are seeing the excerpt which strips HTML tags unless you exclude them from being stripped (see the excerpt options) if that's not the case, you need to explain it better and give the url

That solved it! Thank you. Simply put image tag in options and image appeared.