Well see, here you have good proof that chatGPT isn't actually "the best knowledge retrieving tool at the moment". ChatGPT (and every other LLM) suuuucks at complicated math, because these text extruders don't reason. Seriously, try out some more complicated math problems. I think you'll find chatGPT gets most of them wrong, and in infuriating ways that make very little sense.

I don't disagree that we need better math instruction for students. I've been saying this since I was a student. But using chatGPT being horrible at math as evidence of this is, well, ridiculous, frankly. ChatGPT's performance isn't based on how well your average high schooler understands something, and I don't know why you're trying to tie those two very different things together.

While 3D geometry is more difficult for me than 2D, I could almost immediately tell that the answer is no, there are infinitely many points H that satisfy this. The reason it's unintuitive is that our intuition about what "perpendicular" means comes from 2D and poorly translates to 3D.

The most intuitive explanation I can muster is this: imagine all possible planes that pass through both A and P. It should be obvious that there are infinitely many of them (I visualize it as a plane "rotating" around the AP axis). Each of these planes intersects the given plane since it passes through A. Think of the intersection line. It never passes through P (unless P is on the plane), so it is always possible to draw a perpendicular line from P to that intersection line. With one exception (when the perpendicular line falls on the A point), the point where the perpendicular falls satisfies the conditions for H. (I think all such points actually form a circle with AP' as the diameter, where P' is the parallel projection of P to the given plane, but I'm not 100% sure)