Yesterday I wrote about a problem that Mary Bourassa posted on twitter:
Actually, she wrote about it, too!
I found this problem interesting for a variety of reasons:
(i) It is a nice way to review the law of sines and cosines,
(ii) The relationship between the two triangles that arise in the problem is pretty neat, and gives a nice introduction to the extended law of sines, and
(iii) It shows the importance of checking your work!
Seems like a really nice way to get a good conversation going with a trig class.
Today Christopher Danielson shared this Desmos program:
I’m less sure what to make of this problem. At first I thought maybe it would be interesting to talk about why the parabola and the circle cross the axis in the same spot, but I’m not confident that would be all that interesting. Maybe I’m missing the point, but I don’t see the connection between the circle and the parabola as something that’s going to grab the attention of students.
It made me wonder why I found the first problem more interesting than the second one. Personal taste? Missing the point of the 2nd one? Something else?
In a way both problems are fairly arbitrary – a triangle with essentially random sides and angles, and a pretty random parabola and circle.
Also, what would probably seem to students like a minor change to the 2nd problem – changing it to $y^2 = x^3 + ax + b$ lands you in the field of elliptic curves, which is one of the most interesting fields of math today. So, again, I don’t think the arbitrary nature of the problem is why it didn’t grab me.
Anyway, if I was going to talk about circles and parabolas, I think the problem illustrated on the cover of this book is a fascinating one, and would probably make a neat Desmos activity, too:
The problem goes like this – draw a circle whose diameter is formed by connecting the points (0,1) and (a,b). Now graph the quadratic equation 
Mike's Math Page 
I suppose that to be really interesting a problem has to hit this sweet spot of looking like it should be straightforward to do, but not be so straightforward that it’s instantaneous.
Problems about triangles and circles probably have an edge on problems with parabolas. Everyone knows those shapes, while you don’t even hear the word ‘parabola’ until you get into middle school and have pre-algebra dropped on your head.