Tangent Circle in a Parallelogram for Problem 5088

Completed figures

Parallelogram ABCD with BC equal to BD and M the midpoint of CD. E, F, and G are the perpendicular feet from M to the lines BD, AD, and BC. The circle centered at M passes through E, F, G, and K; AK is tangent at K, and N is the intersection of EG and BM. — The circle centered at M and the auxiliary points used in the proof.
Complete source

Tangent-circle configuration

% Editable source for public/diagrams/crux-5088-tangent-circle.svg.
% This sanitized drawing contains no source problem text or contact information.
% Accent target: none
% Publication Profile: scale 1.0; coordinates compressed for body-sized labels.
% 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(0,0){A}
  % A uniform 0.8 image of the exact 3-4-5 seed configuration.
  \tkzDefPoint(1.6,-3.2){B}
  \tkzDefPoint(5.6,-3.2){C}
  \tkzDefPoint(4.0,0){D}

  \tkzDefMidPoint(C,D)\tkzGetPoint{M}

  \tkzDefLine[perpendicular=through M](B,D)
  \tkzGetPoint{MBD}
  \tkzInterLL(M,MBD)(B,D)\tkzGetPoint{E}
  \tkzDefLine[perpendicular=through M](A,D)
  \tkzGetPoint{MAD}
  \tkzInterLL(M,MAD)(A,D)\tkzGetPoint{F}
  \tkzDefLine[perpendicular=through M](B,C)
  \tkzGetPoint{MBC}
  \tkzInterLL(M,MBC)(B,C)\tkzGetPoint{G}

  \tkzInterLL(E,G)(B,M)\tkzGetPoint{N}
  \tkzDefLine[tangent from=A](M,F)\tkzGetPoints{X}{K}

  \tkzDrawCircle[eg main](M,F)
  \tkzDrawSegments[eg main](A,B B,C C,D D,A B,D A,K B,K K,E)
  \tkzDrawSegments[eg support dashed](F,G B,M E,M F,K E,G K,M)

  \tkzMarkRightAngle[eg mark,size=0.18](M,F,D)
  \tkzMarkRightAngle[eg mark,size=0.18](M,E,D)
  \tkzMarkRightAngle[eg mark,size=0.18](M,G,C)
  \tkzMarkRightAngle[eg mark,size=0.18](A,K,M)
  \tkzMarkSegment[color=egNavy,line width=0.50pt,mark=|](B,C)
  \tkzMarkSegment[color=egNavy,line width=0.50pt,mark=|](B,D)
  \tkzMarkSegment[color=egNavy,line width=0.50pt,mark=||](C,M)
  \tkzMarkSegment[color=egNavy,line width=0.50pt,mark=||](M,D)

  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](A,B,C,D,E,F,G,M,N,K)
  \tkzLabelPoints[above left](A)
  \tkzLabelPoints[below left](B)
  \tkzLabelPoints[below right](C)
  \tkzLabelPoints[above](D)
  \tkzLabelPoints[left](E)
  \tkzLabelPoints[above right](F)
  \tkzLabelPoints[below](G)
  \tkzLabelPoints[right](M)
  \tkzLabelPoints[above left](N)
  \tkzLabelPoints[below,yshift=2pt](K)
  \egUseStandardPlate
\end{tikzpicture}
\end{document}
Read the article

Read Problem 5088 and the proof