A geometry problem involving geometric mean $ABCD$ is a quadrilateral inscribed in a circle of center $O$. Let $BD$ bisect $OC$ perpendicularly. $P$ is a point on the diagonal $AC$ such that $PC=OC$. $BP$ cuts $AD$ at $E$ and the circle circumscribing $ABCD$ at $F$.

Prove that $PF$ is the geometric mean of $EF$ and $BF$.

This is a coordinate-based solution. Without loss of generality, you can choose your coordinate system such that the circle $\bigcirc ABCD$ becomes the unit circle and $C$ has coordinates $(1,0)$. Then, in order to simplify the formulation, we start by choosing rational coordinates for point $P$:
$$P = (2t^2,2t)/(t^2+1)$$
The coordinates are chosen in such a way that they describe any point on the circle $(x-1)^2+y^2=1$ except for the point $(2,0)$ (which corresponds to $t\to\infty$). For any point $P$ on that circle you can compute $t=y/(2-x)$.
From these coordinates, everything else can be computed:
\begin{align*}
O&=(0,0) \\
C&=(1,0) \\
\bigcirc ABCD&:\;\;x^2+y^2=1 \\
\bigcirc OBD&:\;\;(x-1)^2+y^2=1 \\
B&=(1,-\sqrt3)/2 \\
D&=(1,\sqrt3)/2 \\
P&=(2t^2,2t)/(t^2+1) \\
CP&:\;\;2t\,x+(1-t^2)\,y=2t \\
BP&:\;\;(t+\sqrt3)\,x+(1-\sqrt3t)\,y=2t \\
A&=(-t^4 + 6t^2 - 1, -4t^3 + 4t)/(t^4 + 2t^2 + 1) \\
AD&:\;\;(1-t^2-2\sqrt3t)\,x+(\sqrt3t^2 - 2t - \sqrt3)y = t^2 - 2\sqrt3t - 1 \\
E&=(-3\sqrt3t^3 + 11t^2 + \sqrt3t - 1, -3t^3 - 3\sqrt3t^2 + 9t + \sqrt3)/(4t^2 + 4) \\
F&=(t^2 + 2\sqrt3t - 1, -\sqrt3t^2 + 2t + \sqrt3)/(2t^2 + 2) \\
\lVert P-F\rVert^2&=(3t^2 - 2\sqrt3t + 1)/(t^2 + 1) \\
\lVert E-F\rVert^2&=(9t^4 - 12\sqrt3t^3 + 18t^2 - 4\sqrt3t + 1)/(4t^2 + 4) \\
\lVert B-F\rVert^2&=4/(t^2 + 1) \\
\lVert P-F\rVert^4 &= \lVert E-F\rVert^2\cdot\lVert B-F\rVert^2
= (\sqrt3t - 1)^4/(t^2+1)^2
\end{align*}
I did all the computation using projective geometry and homogeneous coordinates. So the things I write as $(x,y)/z$ were internally just homogeneous coordinate vectors $(x,y,z)$ in my computation. And of course I used a computer algebra system (sage in this case) to handle all these polynomials.
