Here is one that was left hanging a few days ago. The parabola below has vertex V and focus F. Points A and P lie on the axis. Tangents AQ and PQ meet at Q.

As though it were common knowledge, I stated that FQ was the geometric mean of FA and FP. When challenged on that, I could not remember where I had learned it, and I did not find it in any of my resources. I have since proved it. That is what this challenge is.

Prove: FQ² = (FA)(FP)

I would prefer a deductive geometry proof, but I suppose an analytic proof would be acceptable if it is generalized.

I see, Indica, you proved it twice. I prefer your second proof over my own.

Corrections on the analytic proof:

VP = s²

VA = t²

By similarity of all parabolas, the analytic proof is generalized.

I will leave it open for a couple of days in case someone else cares to check in.