TITLE: Concyclicity Proof
QUESTION [1 upvotes]: $A,B,C$ are three collinear points and $P$ is a point not in the line $AB$: $AFE,BFD$ and $CED$ are perpendiculars to $PA,PB,PC$ respectively. Prove that $P,D,E,F$ are concyclic.
I attempted to do this using angle chasing. But I can't really understand why the lines are being defined by 3 points, since in my diagrams, this is simply not happening. So I need some help with this.

REPLY [0 votes]: Simson line. Line $AB \equiv AC$ is the Simson line for triangle $EFD$ from the point $P$. By the theorem for the Simson line, the point $P$ lies on the circumcircle of $EFD$ if and only if its three orthogonal projections onto the extended edges of the triangle $EFD$ are collinear, which is the case here.
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