Power and Midpoint Symmetry in BMO1 2025

Completed figure

Acute triangle ABC with M the midpoint of BC. Two circles through M are tangent to AB at B and AC at C and meet again at D. The circumcircle of ABCD meets line DM again at H, the reflection of A across the perpendicular bisector of BC. — The auxiliary point H completes the secant through M and makes the midpoint reflection visible.
Complete source

BMO1 2025 power-and-symmetry configuration

% Figure for a Problem article
% Article: A Solution to BMO1 2025, Problem 4
% Source: British Mathematical Olympiad, BMO1 2025, Problem 4
% Site: Elementary Geometry
% Website: https://elementarygeometry.org/
% Drawing tool: tkz-euclide
% Output format: SVG
% Standard plate: 5:4
% Last verified: 2026-09-06
% Accent target: none
%
% Editable source for public/diagrams/bmo1-2025-power-symmetry.svg.
% The production site serves the committed SVG and does not compile TeX.
\documentclass[tikz,border=8pt]{standalone}
\usepackage{tkz-euclide}

\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]
  % A uniform similarity keeps the explicit seed coordinates at one decimal
  % place while fitting the full circle configuration inside the 5:4 plate.
  \tkzDefPoint(0,0){S}
  \tkzDefPoint(0.5,3.2){A0}
  \tkzDefPoint(-3,0){B0}
  \tkzDefPoint(3,0){C0}
  \tkzDefPointBy[homothety=center S ratio 0.7](A0)\tkzGetPoint{A}
  \tkzDefPointBy[homothety=center S ratio 0.7](B0)\tkzGetPoint{B}
  \tkzDefPointBy[homothety=center S ratio 0.7](C0)\tkzGetPoint{C}
  \tkzDefMidPoint(B,C)\tkzGetPoint{M}

  % Each given circle center is the intersection of the appropriate radius
  % line at the tangency point and the perpendicular bisector of its chord.
  \tkzDefMidPoint(B,M)\tkzGetPoint{BMmid}
  \tkzDefLine[orthogonal=through BMmid](B,M)\tkzGetPoint{bBis}
  \tkzDefLine[orthogonal=through B](A,B)\tkzGetPoint{bNormal}
  \tkzInterLL(B,bNormal)(BMmid,bBis)\tkzGetPoint{OB}

  \tkzDefMidPoint(C,M)\tkzGetPoint{CMmid}
  \tkzDefLine[orthogonal=through CMmid](C,M)\tkzGetPoint{cBis}
  \tkzDefLine[orthogonal=through C](A,C)\tkzGetPoint{cNormal}
  \tkzInterLL(C,cNormal)(CMmid,cBis)\tkzGetPoint{OC}

  \tkzInterCC(OB,B)(OC,C)\tkzGetPoints{Mcheck}{D}

  % The auxiliary circumcircle and the second intersection of line DM.
  \tkzDefCircle[circum](A,B,C)\tkzGetPoint{O}
  \coordinate (OmegaRadiusPoint) at (A);
  \tkzInterLC(D,M)(O,OmegaRadiusPoint)\tkzGetPoints{H}{Dcheck}

  \tkzDrawCircles[eg support](OB,B OC,C)
  \tkzDrawCircle[eg support dashed](O,OmegaRadiusPoint)
  \tkzDrawSegment[eg support dashed](A,M)
  \tkzDrawSegments[eg main](A,B B,C C,A D,H)

  \tkzMarkSegments[color=egNavy,line width=0.50pt,mark=|,size=3pt](B,M M,C)
  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](A,B,C,M,D,H)
  \tkzLabelPoints[above](A)
  \tkzLabelPoints[left](B)
  \tkzLabelPoints[right](C)
  \tkzLabelPoints[below left, xshift=-2pt](M)
  \tkzLabelPoints[below](D)
  \tkzLabelPoints[above](H)

  \egUseStandardPlate
\end{tikzpicture}
\end{document}
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Read the problem and solution