{
  "index": "1976-A-1",
  "type": "GEO",
  "tag": [
    "GEO"
  ],
  "difficulty": "",
  "question": "A-1. \\( P \\) is an interior point of the angle whose sides are the rays \\( O A \\) and \\( O B \\). Locate \\( X \\) on \\( O A \\) and \\( Y \\) on \\( O B \\) so that the line segment \\( \\overline{X Y} \\) contains \\( P \\) and so that the product of distances \\( (P X)(P Y) \\) is a minimum.",
  "solution": "A-1.\nLet \\( \\mu \\) be the angle bisector of \\( \\Varangle A O B \\) and \\( \\lambda \\) be the perpendicular to \\( \\mu \\) through \\( P \\). Then the intersections of \\( \\lambda \\) with \\( O A \\) and \\( O B \\) are chosen as \\( X \\) and \\( Y \\) respectively.\n\nThis construction makes \\( O X=O Y \\) and there is a circle \\( \\Gamma \\) tangent to \\( O A \\) at \\( X \\) and to \\( O B \\) at \\( Y \\). Let \\( \\overline{X_{1} Y_{1}} \\) be any other segment containing \\( P \\) with \\( X_{1} \\) on OA and \\( Y_{1} \\) on OB. Let \\( X_{2} \\) and \\( Y_{2} \\) be the intersections of \\( \\bar{X}_{1} Y_{1} \\) with \\( \\Gamma \\). A theorem of Euclidean geometry states that \\( (P X)(P Y)=\\left(P X_{2}\\right)\\left(P Y_{2}\\right) \\). Clearly \\( \\left(P X_{2}\\right)\\left(P Y_{2}\\right) \\) is less than \\( \\left(P X_{1}\\right)\\left(P Y_{1}\\right) \\). Hence \\( (P X)(P Y) \\) is a minimum.\n\nOne can also locate \\( X \\) and \\( Y \\) by saying that \\( (\\pi-\\Varangle A O B) / 2 \\) should be chosen as the measure of \\( \\Varangle O X P \\) or \\( \\Varangle O Y P \\).",
  "vars": [
    "P",
    "X",
    "Y",
    "O",
    "A",
    "B",
    "X_1",
    "Y_1",
    "X_2",
    "Y_2"
  ],
  "params": [
    "\\\\mu",
    "\\\\lambda",
    "\\\\Gamma"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "P": "pivotpoint",
        "X": "firstpoint",
        "Y": "secondpt",
        "O": "originpt",
        "A": "firstraypt",
        "B": "secondray",
        "X_1": "firstauxone",
        "Y_1": "secondaux",
        "X_2": "firstauxtwo",
        "Y_2": "secondauxtwo",
        "\\mu": "bisectorln",
        "\\lambda": "perpsegm",
        "\\Gamma": "tangentcirc"
      },
      "question": "A-1. \\( pivotpoint \\) is an interior point of the angle whose sides are the rays \\( originpt firstraypt \\) and \\( originpt secondray \\). Locate \\( firstpoint \\) on \\( originpt firstraypt \\) and \\( secondpt \\) on \\( originpt secondray \\) so that the line segment \\( \\overline{firstpoint secondpt} \\) contains \\( pivotpoint \\) and so that the product of distances \\( (pivotpoint firstpoint)(pivotpoint secondpt) \\) is a minimum.",
      "solution": "A-1.\nLet \\( bisectorln \\) be the angle bisector of \\( \\Varangle firstraypt originpt secondray \\) and \\( perpsegm \\) be the perpendicular to \\( bisectorln \\) through \\( pivotpoint \\). Then the intersections of \\( perpsegm \\) with \\( originpt firstraypt \\) and \\( originpt secondray \\) are chosen as \\( firstpoint \\) and \\( secondpt \\) respectively.\n\nThis construction makes \\( originpt firstpoint = originpt secondpt \\) and there is a circle \\( tangentcirc \\) tangent to \\( originpt firstraypt \\) at \\( firstpoint \\) and to \\( originpt secondray \\) at \\( secondpt \\). Let \\( \\overline{firstauxone secondaux} \\) be any other segment containing \\( pivotpoint \\) with \\( firstauxone \\) on \\( originpt firstraypt \\) and \\( secondaux \\) on \\( originpt secondray \\). Let \\( firstauxtwo \\) and \\( secondauxtwo \\) be the intersections of \\( \\bar{firstauxone} secondaux \\) with \\( tangentcirc \\). A theorem of Euclidean geometry states that \\( (pivotpoint firstpoint)(pivotpoint secondpt)=\\left(pivotpoint firstauxtwo\\right)\\left(pivotpoint secondauxtwo\\right) \\). Clearly \\( \\left(pivotpoint firstauxtwo\\right)\\left(pivotpoint secondauxtwo\\right) \\) is less than \\( \\left(pivotpoint firstauxone\\right)\\left(pivotpoint secondaux\\right) \\). Hence \\( (pivotpoint firstpoint)(pivotpoint secondpt) \\) is a minimum.\n\nOne can also locate \\( firstpoint \\) and \\( secondpt \\) by saying that \\( (\\pi-\\Varangle firstraypt originpt secondray) / 2 \\) should be chosen as the measure of \\( \\Varangle originpt firstpoint pivotpoint \\) or \\( \\Varangle originpt secondpt pivotpoint \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "P": "junction",
        "X": "harboring",
        "Y": "landscape",
        "O": "focalpoint",
        "A": "sunflower",
        "B": "lighthouse",
        "X_1": "harboringone",
        "Y_1": "landscapeone",
        "X_2": "harboringtwo",
        "Y_2": "landscapetwo",
        "\\mu": "aurorabeam",
        "\\lambda": "sequoialeaf",
        "\\Gamma": "cobaltcrest"
      },
      "question": "A-1. \\( junction \\) is an interior point of the angle whose sides are the rays \\( focalpoint sunflower \\) and \\( focalpoint lighthouse \\). Locate \\( harboring \\) on \\( focalpoint sunflower \\) and \\( landscape \\) on \\( focalpoint lighthouse \\) so that the line segment \\( \\overline{harboring landscape} \\) contains \\( junction \\) and so that the product of distances \\( (junction harboring)(junction landscape) \\) is a minimum.",
      "solution": "A-1.\nLet \\( aurorabeam \\) be the angle bisector of \\( \\Varangle sunflower focalpoint lighthouse \\) and \\( sequoialeaf \\) be the perpendicular to \\( aurorabeam \\) through \\( junction \\). Then the intersections of \\( sequoialeaf \\) with \\( focalpoint sunflower \\) and \\( focalpoint lighthouse \\) are chosen as \\( harboring \\) and \\( landscape \\) respectively.\n\nThis construction makes \\( focalpoint harboring=focalpoint landscape \\) and there is a circle \\( cobaltcrest \\) tangent to \\( focalpoint sunflower \\) at \\( harboring \\) and to \\( focalpoint lighthouse \\) at \\( landscape \\). Let \\( \\overline{harboringone landscapeone} \\) be any other segment containing \\( junction \\) with \\( harboringone \\) on focalpointsunflower and \\( landscapeone \\) on focalpointlighthouse. Let \\( harboringtwo \\) and \\( landscapetwo \\) be the intersections of \\( \\bar{harboringone} landscapeone \\) with \\( cobaltcrest \\). A theorem of Euclidean geometry states that \\( (junction harboring)(junction landscape)=\\left(junction harboringtwo\\right)\\left(junction landscapetwo\\right) \\). Clearly \\( \\left(junction harboringtwo\\right)\\left(junction landscapetwo\\right) \\) is less than \\( \\left(junction harboringone\\right)\\left(junction landscapeone\\right) \\). Hence \\( (junction harboring)(junction landscape) \\) is a minimum.\n\nOne can also locate \\( harboring \\) and \\( landscape \\) by saying that \\( (\\pi-\\Varangle sunflower focalpoint lighthouse) / 2 \\) should be chosen as the measure of \\( \\Varangle focalpoint harboring junction \\) or \\( \\Varangle focalpoint landscape junction \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "P": "externalpoint",
        "X": "distantpoint",
        "Y": "remoteplace",
        "O": "edgepoint",
        "A": "terminusa",
        "B": "terminusb",
        "X_1": "distantone",
        "Y_1": "fartherone",
        "X_2": "distanttwo",
        "Y_2": "farthertwo",
        "\\mu": "skewdirection",
        "\\lambda": "parallelpath",
        "\\Gamma": "linecluster"
      },
      "question": "A-1. \\( externalpoint \\) is an interior point of the angle whose sides are the rays \\( edgepoint terminusa \\) and \\( edgepoint terminusb \\). Locate \\( distantpoint \\) on \\( edgepoint terminusa \\) and \\( remoteplace \\) on \\( edgepoint terminusb \\) so that the line segment \\( \\overline{distantpoint remoteplace} \\) contains \\( externalpoint \\) and so that the product of distances \\( (externalpoint distantpoint)(externalpoint remoteplace) \\) is a minimum.",
      "solution": "A-1.\nLet \\( skewdirection \\) be the angle bisector of \\( \\Varangle terminusa edgepoint terminusb \\) and \\( parallelpath \\) be the perpendicular to \\( skewdirection \\) through \\( externalpoint \\). Then the intersections of \\( parallelpath \\) with \\( edgepoint terminusa \\) and \\( edgepoint terminusb \\) are chosen as \\( distantpoint \\) and \\( remoteplace \\) respectively.\n\nThis construction makes \\( edgepoint distantpoint=edgepoint remoteplace \\) and there is a circle \\( linecluster \\) tangent to \\( edgepoint terminusa \\) at \\( distantpoint \\) and to \\( edgepoint terminusb \\) at \\( remoteplace \\). Let \\( \\overline{distantone fartherone} \\) be any other segment containing \\( externalpoint \\) with \\( distantone \\) on \\( edgepoint terminusa \\) and \\( fartherone \\) on \\( edgepoint terminusb \\). Let \\( distanttwo \\) and \\( farthertwo \\) be the intersections of \\( \\bar{distantone} fartherone \\) with \\( linecluster \\). A theorem of Euclidean geometry states that \\( (externalpoint distantpoint)(externalpoint remoteplace)=\\left(externalpoint distanttwo\\right)\\left(externalpoint farthertwo\\right) \\). Clearly \\( \\left(externalpoint distanttwo\\right)\\left(externalpoint farthertwo\\right) \\) is less than \\( \\left(externalpoint distantone\\right)\\left(externalpoint fartherone\\right) \\). Hence \\( (externalpoint distantpoint)(externalpoint remoteplace) \\) is a minimum."
    },
    "garbled_string": {
      "map": {
        "P": "qzxwvtnp",
        "X": "hjgrksla",
        "Y": "blfmdqre",
        "O": "cpvzthou",
        "A": "nkdweruio",
        "B": "slarbpeq",
        "X_1": "kfgstlyo",
        "Y_1": "vbnmexzu",
        "X_2": "tqpldskj",
        "Y_2": "whrczvga",
        "\\mu": "gdsneiro",
        "\\lambda": "pkrftoal",
        "\\Gamma": "qhmvzcie"
      },
      "question": "A-1. \\( qzxwvtnp \\) is an interior point of the angle whose sides are the rays \\( cpvzthou nkdweruio \\) and \\( cpvzthou slarbpeq \\). Locate \\( hjgrksla \\) on \\( cpvzthou nkdweruio \\) and \\( blfmdqre \\) on \\( cpvzthou slarbpeq \\) so that the line segment \\( \\overline{hjgrksla blfmdqre} \\) contains \\( qzxwvtnp \\) and so that the product of distances \\( (qzxwvtnp hjgrksla)(qzxwvtnp blfmdqre) \\) is a minimum.",
      "solution": "A-1.\nLet \\( gdsneiro \\) be the angle bisector of \\( \\Varangle nkdweruio cpvzthou slarbpeq \\) and \\( pkrftoal \\) be the perpendicular to \\( gdsneiro \\) through \\( qzxwvtnp \\). Then the intersections of \\( pkrftoal \\) with \\( cpvzthou nkdweruio \\) and \\( cpvzthou slarbpeq \\) are chosen as \\( hjgrksla \\) and \\( blfmdqre \\) respectively.\n\nThis construction makes \\( cpvzthou hjgrksla=cpvzthou blfmdqre \\) and there is a circle \\( qhmvzcie \\) tangent to \\( cpvzthou nkdweruio \\) at \\( hjgrksla \\) and to \\( cpvzthou slarbpeq \\) at \\( blfmdqre \\). Let \\( \\overline{kfgstlyo vbnmexzu} \\) be any other segment containing \\( qzxwvtnp \\) with \\( kfgstlyo \\) on cpvzthounkdweruio and \\( vbnmexzu \\) on cpvzthouslarbpeq. Let \\( tqpldskj \\) and \\( whrczvga \\) be the intersections of \\( \\bar{kfgstlyo vbnmexzu} \\) with \\( qhmvzcie \\). A theorem of Euclidean geometry states that \\( (qzxwvtnp hjgrksla)(qzxwvtnp blfmdqre)=\\left(qzxwvtnp tqpldskj\\right)\\left(qzxwvtnp whrczvga\\right) \\). Clearly \\( \\left(qzxwvtnp tqpldskj\\right)\\left(qzxwvtnp whrczvga\\right) \\) is less than \\( \\left(qzxwvtnp kfgstlyo\\right)\\left(qzxwvtnp vbnmexzu\\right) \\). Hence \\( (qzxwvtnp hjgrksla)(qzxwvtnp blfmdqre) \\) is a minimum.\n\nOne can also locate \\( hjgrksla \\) and \\( blfmdqre \\) by saying that \\( (\\pi-\\Varangle nkdweruio cpvzthou slarbpeq) / 2 \\) should be chosen as the measure of \\( \\Varangle cpvzthou hjgrksla qzxwvtnp \\) or \\( \\Varangle cpvzthou blfmdqre qzxwvtnp \\)."
    },
    "kernel_variant": {
      "question": "Let the rays VU and VW share the vertex V and form an angle smaller than 180°.  A point Q is situated strictly inside ∠UVW.  Using straight-edge and compass construct the points M on VU and N on VW such that the segment MN passes through Q and the product |QM| · |QN| is as small as possible.  Describe the construction and prove that it yields the minimum (and that it is the only one that does so).",
      "solution": "Construction\n1.  Draw the internal angle-bisector β of ∠UVW.\n2.  Through Q draw the line ℓ perpendicular to β.\n3.  Let M = ℓ ∩ VU and N = ℓ ∩ VW.\n\nThat completes the straight-edge-and-compass construction (one perpendicular, two intersections).\n\nProof of optimality\n(i)  A circle tangent to the two sides at M and N.\nBecause ℓ ⟂ β, the acute angles that ℓ makes with the two sides are equal; consequently the triangles VQM and VQN are mirror images in β and VM = VN.\n\nThrough M draw the line m perpendicular to VU and let C = m ∩ β.  Reflecting m in β gives a line n that is perpendicular to VW and passes through N, so CN ⟂ VW and CM ⟂ VU.  Hence CM = CN, and the circle Γ with centre C and radius r = CM is tangent to VU at M and to VW at N.\n\n(ii)  Any other segment through Q meets Γ twice.\nLet d be an arbitrary line through Q contained in the interior of ∠UVW, different from ℓ, and put M₁ = d ∩ VU, N₁ = d ∩ VW.  Travelling from Q along d towards VU we first cross Γ (which lies strictly between the sides except at M and N) and then VU.  Denote the first intersection with Γ by M₂; then M₂ lies on QM₁ and QM₂ < QM₁.  Likewise the second intersection of d with Γ, call it N₂, lies on QN₁, giving QN₂ < QN₁.                                   (1)\n\n(iii)  The powers of the point Q with respect to Γ.\nThe line ℓ meets Γ exactly at the two points M and N, so by the power-of-a-point theorem (using signed products, or absolute values if one prefers unsigned lengths)\n  Pow_Γ(Q) = QM · QN.\nSimilarly, the secant d meets Γ in M₂ and N₂, whence\n  Pow_Γ(Q) = QM₂ · QN₂.\nThus\n  QM · QN = QM₂ · QN₂.                            (2)\n\n(iv)  Comparison of products.\nMultiplying the two strict inequalities in (1) gives QM₂·QN₂ < QM₁·QN₁.  Using equality (2) we obtain\n  QM · QN < QM₁ · QN₁.\nBecause the auxiliary line d was arbitrary (d ≠ ℓ), the pair (M, N) constructed above indeed minimises the product |QM|·|QN|.  Moreover, equality can occur only when d coincides with ℓ, so the minimising segment is unique.\n\nHence the described construction is the only straight-edge-and-compass construction for which the line through Q minimises the product of the distances from Q to the chosen points on the two sides of the angle.",
      "_meta": {
        "core_steps": [
          "Draw the bisector of ∠AOB.",
          "Through P draw the line perpendicular to that bisector; its intersections with OA and OB are X and Y (hence OX = OY).",
          "Use X and Y as tangency points of the unique circle Γ tangent to OA and OB.",
          "Apply the Power-of-a-Point theorem: for any secant through P we have (PX)(PY) = (PX₂)(PY₂), where X₂,Y₂ are the intersections with Γ.",
          "Since X₂,Y₂ lie between P and any other endpoints on OA,OB, (PX)(PY) is minimal precisely when X= X₂ and Y=Y₂, i.e. for the constructed segment."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Names/labels of the two rays that form the angle.",
            "original": "OA and OB"
          },
          "slot2": {
            "description": "Equivalent way to specify the critical line through P (e.g. ‘perpendicular to the bisector’ vs. ‘making equal angles with both sides’).",
            "original": "the line through P perpendicular to the angle bisector"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}