Last week I saw a request for additional solutions to this problem:
I wasn’t in much of a writing mood that night, so I made a short video showing two different solutions involving circumscribed circles:
The second solution – noting that the formula for the area of a triangle is ABC / 4R – really requires trigonometry, and is likely to be too advanced for an intro geometry course.
Today, by interesting coincidence, I saw this neat post on Patrick Honner’s blog:
Patrick Honner on working with Circumcircles in Desmos
That post links to a Desmos program about Circumcircles that is hosted here:
Patrick Honner’s Circumcircle program in Desmos
You can see the solution to the original circle / square problem by playing around with the three points of the triangles in Mr. Honner’s program. In fact if you set 
If you know how to use Desmos, you can probably even add in the original square. Even without the square, though, from the picture above you can see that the diameter of the circle is indeed 25. Yay!
Can’t say that I know how to use Desmos at all (same with Geogebra), but examples like the one Patrick Honner published today show me that these tools are really powerful.
Always happy to see these neat little math coincidences with people sharing math online!
Mike's Math Page 