Incircle Inversion for Problem 5031

Completed figures

Triangle ABC with incircle contact points D, E, and F. A prime and C prime are the midpoints of EF and DE; P lies on DE and on the line through A parallel to EF; X is the intersection of A prime C and IP. — The incircle and auxiliary lines used in the inversion solution.
Complete source

Incircle-inversion configuration

% Editable source for public/diagrams/crux-5031-incircle-inversion.svg.
% This sanitized drawing contains no source problem text or contact information.
% Accent target: none
% Publication Profile: scale 1.0.
% The web build does not compile TeX; export and review the SVG separately.
\documentclass[tikz,border=8pt]{standalone}
\usepackage{tkz-euclide}
\usetikzlibrary{intersections,calc,angles,quotes}

\definecolor{egNavy}{HTML}{17243D}
\definecolor{egGray}{HTML}{777671}
\definecolor{egRust}{HTML}{92513F}
\newcommand{\egUseStandardPlateAt}[1]{%
  \pgfresetboundingbox
  \path[use as bounding box]
    ([xshift=-102pt,yshift=-80pt]#1)
    rectangle
    ([xshift=102pt,yshift=80pt]#1);%
}
\newcommand{\egUseStandardPlate}{%
  \coordinate (egPlateCenter) at (current bounding box.center);%
  \egUseStandardPlateAt{egPlateCenter}%
}
\tikzset{
  eg figure/.style={line cap=round,line join=round,every node/.append style={font=\normalsize,text=egNavy}},
  eg main/.style={draw=egNavy,line width=0.75pt},
  eg support/.style={draw=egGray,line width=0.52pt},
  eg support dashed/.style={draw=egGray,line width=0.52pt,dash pattern=on 3pt off 3pt},
  eg accent/.style={draw=egRust,line width=0.75pt},
  eg mark/.style={draw=egNavy,line width=0.50pt}
}

\begin{document}
\begin{tikzpicture}[eg figure,scale=1.0]
  \tkzDefPoint(2,4){A}
  \tkzDefPoint(0,0){B}
  \tkzDefPoint(6,0){C}

  \tkzDefTriangleCenter[in](A,B,C)\tkzGetPoint{I}

  \tkzDefLine[orthogonal=through I](B,C)\tkzGetPoint{DD}
  \tkzInterLL(I,DD)(B,C)\tkzGetPoint{D}

  \tkzDefLine[orthogonal=through I](C,A)\tkzGetPoint{EE}
  \tkzInterLL(I,EE)(C,A)\tkzGetPoint{E}

  \tkzDefLine[orthogonal=through I](A,B)\tkzGetPoint{FF}
  \tkzInterLL(I,FF)(A,B)\tkzGetPoint{F}

  \tkzDefMidPoint(E,F)\tkzGetPoint{A'}
  \tkzDefMidPoint(D,E)\tkzGetPoint{C'}

  \tkzDefLine[parallel=through A](E,F)\tkzGetPoint{PP}
  \tkzInterLL(D,E)(A,PP)\tkzGetPoint{P}

  \tkzInterLL(A',C)(I,P)\tkzGetPoint{X}

  \tkzDrawSegments[eg main](A,B B,C C,A A',C I,P)
  \tkzDrawSegments[eg support](I,A I,C E,F D,P A,P)

  \tkzMarkRightAngle[eg mark,size=0.18](A,A',E)
  \tkzMarkRightAngle[eg mark,size=0.18](A',A,P)
  \tkzMarkRightAngle[eg mark,size=0.18](I,C',P)

  \tkzDefCircle[circum](D,E,F)\tkzGetPoints{cDEF}{r}
  \tkzDrawCircle[eg main](cDEF,r)

  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](A,B,C,D,E,F,I,A',C',P,X)

  \tkzLabelPoints[above](A)
  \tkzLabelPoints[below](B)
  \tkzLabelPoints[below](C)
  \tkzLabelPoints[below](D)
  \tkzLabelPoints[right,yshift=2pt](E)
  \tkzLabelPoints[left](F)
  \tkzLabelPoints[below,xshift=-1pt](I)
  \tkzLabelPoints[right](P)
  \tkzLabelPoints[above left,yshift=-1pt](A')
  \tkzLabelPoints[below,xshift=4pt,yshift=-2pt](C')
  \tkzLabelPoints[right,yshift=2pt](X)
  \egUseStandardPlate
\end{tikzpicture}
\end{document}
Read the article

Read Problem 5031 and the proof