PDF files open with Preview in a new window and download.

We are homeschoolers and my son uses a MacBook to do some of his studies. All last year whenever we clicked on a link in a certain area of his homeschool website, the link opened a pdf file within the same browser. This school year we click on the same links and now my Mac downloads the file and opens it in a new window. I have spoken with the school and they claim they have made no changes to cause the problem. I also tried on a Windows computer and it works fine, just as it did on the Mac previously. Is there any way to go back to the old way where we click on the link and it opens right there on the same page without downloading and opening a new window? I have tried to work with Adobe and Safari thinking it may be a Safari issue, but no luck. This is very annoying and I would appreciate some assistance if possible.

 

I downloaded FireFox and Adobe but the same problem occurs. I cannot seem to resolve this. I have changed every option to see if I can get it fixed but nothing yet. Can anyone else out there offer some solution? Thanks.