{
  "index": "1967-B-1",
  "type": "GEO",
  "tag": [
    "GEO",
    "ALG"
  ],
  "difficulty": "",
  "question": "B-1. Let \\( (A B C D E F) \\) be a hexagon inscribed in a circle of radius \\( \\% \\). Show that if \\( \\overline{A B}=\\bar{C} \\bar{D} \\) \\( =\\overline{E F}=\\varphi \\), then the midpoints of \\( \\overline{B C}, \\overline{D E}, \\overline{F A} \\) are the vertices of an equilateral triangle.",
  "solution": "B-1 Consider the figure in the complex plane with the center of the circle at the origin. We can take \\( A, B, C, D, E, F \\) as complex numbers of absolute value \\( r \\). Furthermore \\( B=A \\omega, D=C \\omega \\) and \\( F=E \\omega \\), where \\( \\omega=\\cos (\\pi / 3)+i \\sin (\\pi / 3) \\). Since \\( \\omega^{3}=1 \\) and \\( \\omega \\neq 1, \\omega^{2}-\\omega+1=0 \\). The mid-points of \\( B C, D E \\) and \\( F A \\) are \\( P=\\frac{1}{2}(A \\omega+C), Q=\\frac{1}{2}(C \\omega+E) \\) and \\( R=\\frac{1}{2}(E \\omega+A) \\). If the segment from \\( Q \\) to \\( R \\) is rotated through \\( \\pi / 3 \\) about \\( Q \\), then \\( R \\) is carried into \\( Q+\\omega(R-Q) \\), which equals \\( P \\). Thus \\( P, Q, R \\) are vertices of an equilateral triangle.",
  "vars": [
    "A",
    "B",
    "C",
    "D",
    "E",
    "F",
    "P",
    "Q",
    "R"
  ],
  "params": [
    "r",
    "\\\\omega"
  ],
  "sci_consts": [
    "i"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "A": "pointa",
        "B": "pointb",
        "C": "pointc",
        "D": "pointd",
        "E": "pointe",
        "F": "pointf",
        "P": "midpointp",
        "Q": "midpointq",
        "R": "midpointr",
        "r": "radius",
        "\\omega": "rootunity"
      },
      "question": "B-1. Let \\( (pointa pointb pointc pointd pointe pointf) \\) be a hexagon inscribed in a circle of radius \\( radius \\). Show that if \\( \\overline{pointa pointb}=\\bar{pointc} \\bar{pointd} \\) \\( =\\overline{pointe pointf}=\\varphi \\), then the midpoints of \\( \\overline{pointb pointc}, \\overline{pointd pointe}, \\overline{pointf pointa} \\) are the vertices of an equilateral triangle.",
      "solution": "B-1 Consider the figure in the complex plane with the center of the circle at the origin. We can take \\( pointa, pointb, pointc, pointd, pointe, pointf \\) as complex numbers of absolute value \\( radius \\). Furthermore \\( pointb = pointa rootunity, pointd = pointc rootunity \\) and \\( pointf = pointe rootunity \\), where \\( rootunity = \\cos (\\pi / 3) + i \\sin (\\pi / 3) \\). Since \\( rootunity^{3} = 1 \\) and \\( rootunity \\neq 1, \\; rootunity^{2} - rootunity + 1 = 0 \\). The mid-points of \\( pointb pointc, pointd pointe \\) and \\( pointf pointa \\) are \\( midpointp = \\tfrac{1}{2}(pointa rootunity + pointc), \\; midpointq = \\tfrac{1}{2}(pointc rootunity + pointe) \\) and \\( midpointr = \\tfrac{1}{2}(pointe rootunity + pointa) \\). If the segment from \\( midpointq \\) to \\( midpointr \\) is rotated through \\( \\pi / 3 \\) about \\( midpointq \\), then \\( midpointr \\) is carried into \\( midpointq + rootunity(midpointr - midpointq) \\), which equals \\( midpointp \\). Thus \\( midpointp, midpointq, midpointr \\) are vertices of an equilateral triangle."
    },
    "descriptive_long_confusing": {
      "map": {
        "A": "satellite",
        "B": "landscape",
        "C": "honeycomb",
        "D": "pineapple",
        "E": "spaceship",
        "F": "groundhog",
        "P": "raincloud",
        "Q": "waterfall",
        "R": "dragonsky",
        "r": "longitude",
        "\\omega": "megaforce"
      },
      "question": "B-1. Let \\( (satellite landscape honeycomb pineapple spaceship groundhog) \\) be a hexagon inscribed in a circle of radius \\( \\% \\). Show that if \\( \\overline{satellite landscape}=\\bar{honeycomb} \\bar{pineapple} =\\overline{spaceship groundhog}=\\varphi \\), then the midpoints of \\( \\overline{landscape honeycomb}, \\overline{pineapple spaceship}, \\overline{groundhog satellite} \\) are the vertices of an equilateral triangle.",
      "solution": "B-1 Consider the figure in the complex plane with the center of the circle at the origin. We can take \\( satellite, landscape, honeycomb, pineapple, spaceship, groundhog \\) as complex numbers of absolute value \\( longitude \\). Furthermore \\( landscape=satellite\\, megaforce,\\; pineapple=honeycomb\\, megaforce \\) and \\( groundhog=spaceship\\, megaforce \\), where \\( megaforce=\\cos (\\pi / 3)+i \\sin (\\pi / 3) \\). Since \\( megaforce^{3}=1 \\) and \\( megaforce \\neq 1,\\; megaforce^{2}-megaforce+1=0 \\). The mid-points of \\( landscape honeycomb,\\; pineapple spaceship \\) and \\( groundhog satellite \\) are \\( raincloud=\\tfrac12(satellite\\, megaforce+honeycomb),\\; waterfall=\\tfrac12(honeycomb\\, megaforce+spaceship) \\) and \\( dragonsky=\\tfrac12(spaceship\\, megaforce+satellite) \\). If the segment from \\( waterfall \\) to \\( dragonsky \\) is rotated through \\( \\pi / 3 \\) about \\( waterfall \\), then \\( dragonsky \\) is carried into \\( waterfall+megaforce(dragonsky-waterfall) \\), which equals \\( raincloud \\). Thus \\( raincloud,\\; waterfall,\\; dragonsky \\) are vertices of an equilateral triangle."
    },
    "descriptive_long_misleading": {
      "map": {
        "A": "voidpoint",
        "B": "blankpoint",
        "C": "nilpoint",
        "D": "nullpoint",
        "E": "vacancypoint",
        "F": "emptiness",
        "P": "nonvertex",
        "Q": "offcenter",
        "R": "deviation",
        "r": "centerpoint",
        "\\omega": "straightness"
      },
      "question": "B-1. Let \\( (voidpoint blankpoint nilpoint nullpoint vacancypoint emptiness) \\) be a hexagon inscribed in a circle of radius \\( centerpoint \\). Show that if \\( \\overline{voidpoint blankpoint}=\\bar{nilpoint} \\bar{nullpoint} \\) \\( =\\overline{vacancypoint emptiness}=\\varphi \\), then the midpoints of \\( \\overline{blankpoint nilpoint}, \\overline{nullpoint vacancypoint}, \\overline{emptiness voidpoint} \\) are the vertices of an equilateral triangle.",
      "solution": "B-1 Consider the figure in the complex plane with the center of the circle at the origin. We can take \\( voidpoint, blankpoint, nilpoint, nullpoint, vacancypoint, emptiness \\) as complex numbers of absolute value \\( centerpoint \\). Furthermore \\( blankpoint=voidpoint\\,straightness, nullpoint=nilpoint\\,straightness \\) and \\( emptiness=vacancypoint\\,straightness \\), where \\( straightness=\\cos (\\pi / 3)+i \\sin (\\pi / 3) \\). Since \\( straightness^{3}=1 \\) and \\( straightness \\neq 1, straightness^{2}-straightness+1=0 \\). The mid-points of \\( blankpoint nilpoint, nullpoint vacancypoint \\) and \\( emptiness voidpoint \\) are \\( nonvertex=\\frac{1}{2}(voidpoint\\,straightness+nilpoint), offcenter=\\frac{1}{2}(nilpoint\\,straightness+vacancypoint) \\) and \\( deviation=\\frac{1}{2}(vacancypoint\\,straightness+voidpoint) \\). If the segment from \\( offcenter \\) to \\( deviation \\) is rotated through \\( \\pi / 3 \\) about \\( offcenter \\), then \\( deviation \\) is carried into \\( offcenter+straightness(deviation-offcenter) \\), which equals \\( nonvertex \\). Thus \\( nonvertex, offcenter, deviation \\) are vertices of an equilateral triangle."
    },
    "garbled_string": {
      "map": {
        "A": "qzxwvtnp",
        "B": "hjgrksla",
        "C": "pmcftwov",
        "D": "nlrkzgye",
        "E": "sdqivjpo",
        "F": "xluqearn",
        "P": "vndoswya",
        "Q": "tkemaghi",
        "R": "yzprlcwm",
        "r": "bsaovmlc",
        "\\omega": "qufydaje"
      },
      "question": "B-1. Let \\( (qzxwvtnp hjgrksla pmcftwov nlrkzgye sdqivjpo xluqearn) \\) be a hexagon inscribed in a circle of radius \\( \\% \\). Show that if \\( \\overline{qzxwvtnp hjgrksla}=\\bar{pmcftwov} \\bar{nlrkzgye} \\) \\( =\\overline{sdqivjpo xluqearn}=\\varphi \\), then the midpoints of \\( \\overline{hjgrksla pmcftwov}, \\overline{nlrkzgye sdqivjpo}, \\overline{xluqearn qzxwvtnp} \\) are the vertices of an equilateral triangle.",
      "solution": "B-1 Consider the figure in the complex plane with the center of the circle at the origin. We can take \\( qzxwvtnp, hjgrksla, pmcftwov, nlrkzgye, sdqivjpo, xluqearn \\) as complex numbers of absolute value \\( bsaovmlc \\). Furthermore \\( hjgrksla=qzxwvtnp qufydaje, nlrkzgye=pmcftwov qufydaje \\) and \\( xluqearn=sdqivjpo qufydaje \\), where \\( qufydaje=\\cos (\\pi / 3)+i \\sin (\\pi / 3) \\). Since \\( qufydaje^{3}=1 \\) and \\( qufydaje \\neq 1, qufydaje^{2}-qufydaje+1=0 \\). The mid-points of \\( hjgrksla pmcftwov, nlrkzgye sdqivjpo \\) and \\( xluqearn qzxwvtnp \\) are \\( vndoswya=\\frac{1}{2}(qzxwvtnp qufydaje+pmcftwov), tkemaghi=\\frac{1}{2}(pmcftwov qufydaje+sdqivjpo) \\) and \\( yzprlcwm=\\frac{1}{2}(sdqivjpo qufydaje+qzxwvtnp) \\). If the segment from \\( tkemaghi \\) to \\( yzprlcwm \\) is rotated through \\( \\pi / 3 \\) about \\( tkemaghi \\), then \\( yzprlcwm \\) is carried into \\( tkemaghi+qufydaje(yzprlcwm-tkemaghi) \\), which equals \\( vndoswya \\). Thus \\( vndoswya, tkemaghi, yzprlcwm \\) are vertices of an equilateral triangle."
    },
    "kernel_variant": {
      "question": "Let $P Q R S T U$ be a convex hexagon inscribed in a circle of radius $\\rho$.  Assume that the three pairwise non-adjacent sides\\[ PQ = RS = TU = \\rho. \\]Show that the mid-points of the remaining sides $QR$, $ST$ and $UP$ form an equilateral triangle.",
      "solution": "Put the circle in the complex plane with its centre at the origin.  Denote the complex affixes of the vertices by\n$$p,q,r,s,t,u\\\\ (|p|=|q|=|r|=|s|=|t|=|u|=\\rho).$$\n\n1.  A chord in a circle equals the radius exactly when the corresponding central angle is $60^{\\circ}$.  Consequently there is a $60^{\\circ}$ rotation taking every vertex of a designated unit side to its neighbour on that side.  Choose the negative (clockwise) $60^{\\circ}$ rotation\n$$\\zeta = \\cos(-\\pi/3)+i\\sin(-\\pi/3)=e^{-i\\pi/3},\\qquad \\zeta^2-\\zeta+1=0,\\;|\\zeta|=1,$$\nso that\n$$q=p\\zeta,\\qquad s=r\\zeta,\\qquad u=t\\zeta.\\tag{1}$$\n\n2.  Let\n$$M=\\frac{q+r}{2},\\qquad N=\\frac{s+t}{2},\\qquad L=\\frac{u+p}{2}$$\nbe the mid-points of $QR,ST,UP$, respectively.\n\n3.  Rotate the segment $NL$ about $N$ through $-60^{\\circ}$, i.e. multiply the vector $L-N$ by $\\zeta$:\n\\begin{align*}\nN+\\zeta(L-N)\n&=\\frac{s+t}{2}+\\zeta\\Bigl(\\frac{u+p}{2}-\\frac{s+t}{2}\\Bigr)\\\\[2mm]\n&=\\frac{r\\zeta+t}{2}+\\zeta\\,\\frac{t\\zeta+p-r\\zeta-t}{2}&&\\text{(using (1))}\\\\[2mm]\n&=\\frac{r\\zeta+t+\\zeta^2(t-r)+\\zeta p-\\zeta t}{2}\\\\[2mm]\n&=\\frac{r(\\zeta-\\zeta^2)+t(1+\\zeta^2-\\zeta)+\\zeta p}{2}.\\tag{2}\n\\end{align*}\nBecause $\\zeta^2-\\zeta+1=0$, we have $\\zeta-\\zeta^2=1$ and $1+\\zeta^2-\\zeta=0$.  Thus (2) simplifies to\n$$N+\\zeta(L-N)=\\frac{r+\\zeta p}{2}=\\frac{r+q}{2}=M.$$  \n\n4.  Hence the $-60^{\\circ}$ rotation about $N$ carries $L$ to $M$, so $\\triangle MNL$ is equilateral.  Therefore the mid-points of $QR$, $ST$ and $UP$ are indeed the vertices of an equilateral triangle, as required.\n\n$\\boxed{}$",
      "_meta": {
        "core_steps": [
          "Model the circle in the complex plane with center 0 (vertices have |z| = r).",
          "Equal chord = radius ⇒ central angle 60°, so B = A·ω, D = C·ω, F = E·ω with ω ² − ω + 1 = 0.",
          "Write midpoints P,Q,R of BC, DE, FA as ½(Aω+C), ½(Cω+E), ½(Eω+A).",
          "Rotate segment QR about Q by factor ω: Q + ω(R−Q) = P.",
          "Hence P,Q,R are the vertices of an equilateral triangle."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Symbol/name chosen for the circle’s radius",
            "original": "r"
          },
          "slot2": {
            "description": "Exact root of unity chosen (ω or its conjugate) that satisfies ω² − ω + 1 = 0",
            "original": "ω = cos(π/3) + i sin(π/3)"
          },
          "slot3": {
            "description": "Which three pairwise-nonadjacent sides are declared equal",
            "original": "AB = CD = EF"
          },
          "slot4": {
            "description": "Which complementary edges’ midpoints are considered",
            "original": "midpoints of BC, DE, FA"
          },
          "slot5": {
            "description": "Labeling/start point of the hexagon (cyclic relabeling)",
            "original": "(A, B, C, D, E, F)"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}