\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e \\(1\\) \u70b9 P \\(\\left( \\dfrac{1}{2} , \\dfrac{1}{4} \\right)\\) \u3092\u3068\u308b.\r\n\u653e\u7269\u7dda \\(y = x^2\\) \u4e0a\u306e \\(2\\) \u70b9 Q \\(( \\alpha , \\alpha^2 )\\) , R \\(( \\beta , \\beta^2 )\\) \u3092,\r\n\\(3\\) \u70b9 P , Q , R \u304c QR \u3092\u5e95\u8fba\u3068\u3059\u308b\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u3092\u306a\u3059\u3088\u3046\u306b\u52d5\u304b\u3059\u3068\u304d,\r\n\u25b3PQR \u306e\u91cd\u5fc3 G \\(( X , Y )\\) \u306e\u8ecc\u8de1\u3092\u6c42\u3081\u3088.<\/p>\r\n
\u70b9 G \u306f \u25b3PQR \u306e\u91cd\u5fc3\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nX = \\dfrac{\\alpha +\\beta +\\dfrac{1}{2}}{3} , \\quad Y = \\dfrac{\\alpha^2 +\\beta^2 +\\dfrac{1}{4}}{3} \\\\\r\n\\text{\u2234} \\quad \\alpha +\\beta = 3X -\\dfrac{1}{2} , \\quad \\alpha^2 +\\beta^2 = 3Y -\\dfrac{1}{4} \\quad ... [1]\r\n\\end{align}\\]\r\n\\(\\text{PQ} = \\text{PR}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\left( \\alpha -\\dfrac{1}{2} \\right)^2 +\\left( \\alpha^2 -\\dfrac{1}{4} \\right)^2 = \\left( \\beta -\\dfrac{1}{2} \\right)^2 +\\left( \\beta^2 -\\dfrac{1}{4} \\right)^2 \\quad ... [2] \\\\\r\n( \\alpha -\\beta ) ( \\alpha +\\beta -1 ) +( \\alpha^2 -\\beta^2 ) \\left( \\alpha^2 +\\beta^2 -\\dfrac{1}{2} \\right) = 0 \\\\\r\n( \\alpha +\\beta -1 ) + ( \\alpha +\\beta ) \\left( \\alpha^2 +\\beta^2 -\\dfrac{1}{2} \\right) = 0 \\quad ( \\ \\text{\u2235} \\ \\alpha -\\beta \\neq 0 \\ ) \\\\\r\n( \\alpha +\\beta ) \\left( \\alpha^2 +\\beta^2 +\\dfrac{1}{2} \\right) -1= 0 \\quad ... [3]\r\n\\end{align}\\]\r\n[1] \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\left( 3X -\\dfrac{1}{2} \\right) \\left( 3Y +\\dfrac{1}{4} \\right) & = 1 \\\\\r\n\\text{\u2234} \\quad \\left( X -\\dfrac{1}{6} \\right) \\left( Y +\\dfrac{1}{12} \\right) & = \\dfrac{1}{9}\r\n\\end{align}\\]\r\n\u7d9a\u3044\u3066, \\(X\\) \u306e\u53d6\u308a\u5f97\u308b\u5024\u306e\u7bc4\u56f2\u306b\u3064\u3044\u3066\u8003\u3048\u308b.
\r\n\u6761\u4ef6\u3088\u308a, \\(\\alpha = \\beta\\) \u3068\u306a\u308b\u3053\u3068\u306f\u306a\u3044\u306e\u3067, \\(\\alpha \\neq \\beta\\) \u3068\u306a\u308b\u305f\u3081\u306e \\(X\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
\r\n\\(s = \\alpha +\\beta\\) , \\(t = \\alpha^2 +\\beta^2\\) \u3068\u304a\u304f.\r\n[1] \u3088\u308a\r\n\\[\r\n\\alpha \\beta = \\dfrac{s^2 -t}{2}\r\n\\]\r\n\u306a\u306e\u3067, \\(\\alpha , \\beta\\) \u306f \\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(z^2 -sz +\\dfrac{s^2 -t}{2} = 0\\) \u306e\u7570\u306a\u308b \\(2\\) \u89e3\u306a\u306e\u3067,\r\n\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nD = s^2 -4 \\cdot \\dfrac{s^2 -t}{2} & =-s^2 +2t \\gt 0 \\\\\r\n\\text{\u2234} \\quad s^2 & \\lt 2t \\quad ... [4]\r\n\\end{align}\\]\r\n[3] \u3088\u308a\r\n\\[\\begin{align}\r\ns \\left( t +\\dfrac{1}{2} \\right) & = 1 \\\\\r\n\\text{\u2234} \\quad t & = \\dfrac{1}{s} -\\dfrac{1}{2}\r\n\\end{align}\\]\r\n\\(t \\gt 0\\) \u306a\u306e\u3067, \\(0 \\lt s \\lt 2\\) .
\r\n[4] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\ns^2 \\lt \\dfrac{2}{s} -1 \\\\\r\ns^3 +s -2 \\lt 0 \\\\\r\n( s-1 )( s^2 +s +2 ) \\lt 0 \\\\\r\n\\text{\u2234} \\quad 0 \\lt s \\lt 1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n0 \\lt 3X -\\dfrac{1}{2} \\lt 1 \\\\\r\n\\text{\u2234} \\quad \\dfrac{1}{6} \\lt X \\lt \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a\u6c42\u3081\u308b\u8ecc\u8de1\u306f\r\n\\[\r\n\\underline{\\text{\u53cc\u66f2\u7dda} \\ : \\ \\left( x -\\dfrac{1}{6} \\right) \\left( y +\\dfrac{1}{12} \\right) = \\dfrac{1}{9} \\quad \\left( \\dfrac{1}{6} \\lt x \\lt \\dfrac{1}{2} \\right)}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e \\(1\\) \u70b9 P \\(\\left( \\dfrac{1}{2} , \\dfrac{1}{4} \\right)\\) \u3092\u3068\u308b. \u653e\u7269\u7dda \\(y = x^2\\) \u4e0a\u306e \\(2\\) \u70b9 Q \\(( \\alpha , … \u7d9a\u304d\u3092\u8aad\u3080