第２６回数学的な応募問題

　いつもの通りにパソコンのプログラム（ＶＩＳＵＡＬ　ＢＡＳＩＣ）で解いてみました．

問題１：単位円で考えます．Ｍ(０,−１)とし，ｘ軸とＡＢが平行であるとします．Ａ，Ｓ，Ｒの位置を乱数で決め，それぞれの場合のＢ，Ｐ，Ｑの位置を求めます．さらに∠ＰＳＲ＋∠ＰＱＲを計算し，それが１８０°に等しいことが分かれば，四角形ＰＱＲＳが円に内接することが分かります．この試行を１０００回（＝kaisuu）繰り返し，どんな場合でも円に内接することを示します．

問題２：単位円で考えます．ｘ軸とＡＢが平行であるとします．Ａ，Ｐの位置を乱数で決め，それぞれの場合のＢ，Ｃ，Ｄ，Ｑ，Ｒの位置を求めます．さらにＰＱ／ＰＲを計算し，それが１に等しいことが分かれば，△ＰＱＲが二等辺三角形であることが分かります．この試行を１０００回（＝kaisuu）繰り返し，どんな場合でも二等辺三角形であることを示します．

問題３：実はパソコンのプログラムで解こうとしたところ，あっという間に解答が出てしまったので，プログラムを作る気力がなくなってしまいました．

　円の中心をＯとします．∠ＡＤＥ＝４５°なので∠ＡＯＥ＝９０°であり，ＯＡ＝ＯＥであるから，∠ＯＡＥ＝４５°となるので，∠ＤＡＯ＝∠ＤＡＥ−∠ＯＡＥ＝１５°です．故に∠ＤＦＧ＝１５°＋９０°＝１０５°となります．アルハゼンの定理を使いませんでした．申し訳有りません．

　しかしこれでは，私の気がすまないので，グラフ作成ソフトＧＲＡＰＥＳ（友田勝久氏作成，インターネットからダウンロード可能，フリーソフトです）を使って，解いてみました．答の点をマウスでなぞって，指示通りｘ座標を１００倍すると，答の１０５°が出て来ます．

　この図は次のように作成しました．単位円で考えます．Ａ(０,−１)，Ｅ(１,０)，Ｄ(cos１２０°,sin１２０°)としたとき，直線ＡＤとｘ軸のなす角が答になります．

問題４：単位円で考えます．中心をＯとするとき，ｘ軸とＯＤのなす角ｔＤを，９０°＜ｔＤ＜１８０とします．それぞれのｔＤの値に対して，ＤＦ／ＧＥ＝４となるＡの位置を求めて，相似比を計算して，ＤＦ＝８となるときのＡＦの長さを求めます．このプログラムにより，答は常にＡＦ＝４（�p）となることが分かります．

　乱数がからむ問題１と２のプログラムは見ていて面白いものです．毎回動きが違うので，見ていて思わず「がんばれ！」と言ってしまいます．

問題１プログラム

Option Explicit

Const pi = 3.14159265359

Const r = 1

Sub Form_Load()

Dim waku As Double: waku = 1.5 * r

Picture1.Scale (-waku, waku)-(waku, -waku)

End Sub

Sub Command1_Click()

'm26_1

'　弧ＡＭの長さと弧ＢＭの長さが等しいとき，

'四角形ＰＱＲＳは円に内接していることを証明してください．

Randomize Timer

Dim j, iro0, iro1, iro2, kaisuu As Integer

iro0 = 50: iro1 = 100: iro2 = 200: kaisuu = 1000

Dim wt, wtt As Long: wtt = 150000

Dim w, tA, tB, tS, tR As Double: w = pi / 180

Dim xM, yM, xA, yA, xB, yB, xS, yS, xR, yR, xP, yP, xQ, yQ As Double

Dim dA, dS, dR As Double

Dim PS, SR, PR, PQ, QR, kakuPSR, kakuPQR, a, b, c, d, p, q, sa As Double

Dim xO, yO, rr As Double: sa = 15

xM = 0: yM = -r

tA = ((270 - sa) - (90 + 1.5 * sa)) * Rnd + (90 + 1.5 * sa)

tB = 180 - tA

tS = ((tA - sa) - (tB + 2 * sa)) * Rnd + (tB + 2 * sa)

tR = ((tS - sa) - (tB + sa)) * Rnd + (tB + sa)

dA = 2 * (1 - 2 * Rnd): dS = 2 * (1 - 2 * Rnd)

dR = 2 * (1 - 2 * Rnd)

For j = 0 To kaisuu

If j > 0 Then

dA = inf(dA + 1 - 2 * Rnd): tA = tA + dA

If tA > 270 - sa Then

tA = 270 - sa: dA = -dA

ElseIf tA < 90 + 2 * sa Then

tA = 90 + 2 * sa: dA = -dA

End If

tB = 180 - tA

dS = inf(dS + 1 - 2 * Rnd): tS = tS + dS

If tA - tS < sa Then

tS = tA - sa: dS = -dS

ElseIf tS < tB + 2 * sa Then

tS = tB + 2 * sa: dS = -dS

End If

dR = inf(dR + 1 - 2 * Rnd): tR = tR + dR

If tS - tR < sa Then tR = tS - sa: dR = -dR

If tR - tB < sa Then tR = tB + sa: dR = -dR

End If

xA = r * Cos(tA * w): yA = r * Sin(tA * w)

xB = r * Cos(tB * w): yB = r * Sin(tB * w)

xS = r * Cos(tS * w): yS = r * Sin(tS * w)

xR = r * Cos(tR * w): yR = r * Sin(tR * w)

If xS <> xM Then

a = (yS - yM) / (xS - xM): b = yM - xM * a

xP = (yA - b) / a: yP = yA

Else

xP = 0: yP = r

End If

If xR <> xM Then

a = (yR - yM) / (xR - xM): b = yM - xM * a

xQ = (yB - b) / a: yQ = yB

Else

xQ = 0: yQ = r

End If

PR = Sqr((xR - xP) * (xR - xP) + (yR - yP) * (yR - yP))

PS = Sqr((xS - xP) * (xS - xP) + (yS - yP) * (yS - yP))

SR = Sqr((xR - xS) * (xR - xS) + (yR - yS) * (yR - yS))

kakuPSR = kaku(PS, SR, PR)

PQ = Sqr((xQ - xP) * (xQ - xP) + (yQ - yP) * (yQ - yP))

QR = Sqr((xR - xQ) * (xR - xQ) + (yR - yQ) * (yR - yQ))

kakuPQR = kaku(PQ, QR, PR)

a = xP - xQ: b = yP - yQ: c = xP - xR: d = yP - yR

p = (xP * xP - xQ * xQ + yP * yP - yQ * yQ) * 0.5

q = (xP * xP - xR * xR + yP * yP - yR * yR) * 0.5

xO = (d * p - b * q) / (a * d - b * c)

yO = (a * q - c * p) / (a * d - b * c)

rr = Sqr((xP - xO) * (xP - xO) + (yP - yO) * (yP - yO))

Picture2.Cls

Picture3.Cls: Picture3.Print "残り"; kaisuu - j; "回"

Picture1.Cls

Picture1.Circle (0, 0), r, iro0

Picture1.Line (xA, yA)-(xB, yB), iro1

Picture1.Line (xM, yM)-(xS, yS), iro1

Picture1.Line -(xR, yR), iro1: Picture1.Line -(xM, yM), iro1

Picture1.Line (xP, yP)-(xR, yR), iro1

Picture1.Circle (xO, yO), rr, iro2

Picture1.CurrentX = xA: Picture1.CurrentY = yA

Picture1.Print "A"

Picture1.CurrentX = xB: Picture1.CurrentY = yB

Picture1.Print "B"

Picture1.CurrentX = xM: Picture1.CurrentY = yM

Picture1.Print "M"

Picture1.CurrentX = xS: Picture1.CurrentY = yS

Picture1.Print "S"

Picture1.CurrentX = xR: Picture1.CurrentY = yR

Picture1.Print "R"

Picture1.CurrentX = xP: Picture1.CurrentY = yP

Picture1.Print "P"

Picture1.CurrentX = xQ: Picture1.CurrentY = yQ

Picture1.Print "Q"

For wt = 0 To wtt: Next

Next

End Sub

Private Sub Command2_Click()

End

End Sub

Private Function shishagonyuu(ByVal x As Double) As Double

Dim keta As Long: keta = 1000000

shishagonyuu = Int(x * keta + 0.5) / keta

End Function

Private Function kaku(ByVal a As Double, ByVal b As Double, ByVal c As Double) As Double

Dim cs As Double: cs = (a * a + b * b - c * c) / a / b * 0.5

If cs = 0 Then

kaku = 90

Else

kaku = Atn(Sqr(1 / cs / cs - 1)) / pi * 180

If cs < 0 Then kaku = 180 - kaku

End If

End Function

Private Function inf(ByVal x As Double) As Double

If Abs(x) > 2 Then inf = Sgn(x) * 2 Else inf = x

End Function

問題２プログラム

Option Explicit

Const pi = 3.14159265359

Const r = 1

Sub Form_Load()

Dim waku As Double: waku = 1.5 * r

Picture1.Scale (-waku, waku)-(waku, -waku)

End Sub

Sub Command1_Click()

'm26_2

'　弧ＡＰの長さと弧ＡＣの長さが等しく，

'また弧ＢＰの長さと弧ＢＤの長さが等しいとき，

'三角形ＰＱＲは二等辺三角形であることを証明してください．

Randomize Timer

Dim j, iro0, iro1, kaisuu As Integer

iro0 = 50: iro1 = 200: kaisuu = 1000

Dim wt, wtt As Long: wtt = 200000

Dim w, tA, tB, tP, tC, tD As Double: w = pi / 180

Dim xA, yA, xB, yB, xC, yC, xD, yD, xP, yP, xQ, yQ, xR, yR As Double

Dim dA, dP, PQ, PR, a, b, sa As Double: sa = 10

tA = ((180 - sa) - (90 + sa)) * Rnd + (90 + sa)

tB = 180 - tA

tP = ((tA - sa) - (tB + sa)) * Rnd + (tB + sa)

dA = 2 * (1 - 2 * Rnd): dP = 2 * (1 - 2 * Rnd)

For j = 0 To kaisuu

If j > 0 Then

dA = inf(dA + 1 - 2 * Rnd): tA = tA + dA

If tA > 180 - sa Then

tA = 180 - sa: dA = -dA

ElseIf tA < 90 + sa Then

tA = 90 + sa: dA = -dA

End If

tB = 180 - tA

dP = inf(dP + 1 - 2 * Rnd): tP = tP + dP

If tA - tP < sa Then

tP = tA - sa: dP = -dP

ElseIf tP < tB + sa Then

tP = tB + sa: dP = -dP

End If

End If

tC = 2 * tA - tP: tD = 2 * tB - tP

xA = r * Cos(tA * w): yA = r * Sin(tA * w)

xB = r * Cos(tB * w): yB = r * Sin(tB * w)

xC = r * Cos(tC * w): yC = r * Sin(tC * w)

xD = r * Cos(tD * w): yD = r * Sin(tD * w)

xP = r * Cos(tP * w): yP = r * Sin(tP * w)

a = (yP - yC) / (xP - xC): b = yP - xP * a

xQ = (yA - b) / a: yQ = yA

a = (yP - yD) / (xP - xD): b = yP - xP * a

xR = (yB - b) / a: yR = yB

PQ = Sqr((xQ - xP) * (xQ - xP) + (yQ - yP) * (yQ - yP))

PR = Sqr((xR - xP) * (xR - xP) + (yR - yP) * (yR - yP))

Picture2.Cls

Picture2.Print "PQ / PR="; shishagonyuu(PQ / PR)

Picture3.Cls: Picture3.Print "残り"; kaisuu - j; "回"

Picture1.Cls

Picture1.Circle (0, 0), r, iro0

Picture1.Line (xA, yA)-(xB, yB), iro1

Picture1.Line (xC, yC)-(xP, yP), iro1

Picture1.Line -(xD, yD), iro1

Picture1.CurrentX = xA: Picture1.CurrentY = yA

Picture1.Print "A"

Picture1.CurrentX = xB: Picture1.CurrentY = yB

Picture1.Print "B"

Picture1.CurrentX = xC: Picture1.CurrentY = yC

Picture1.Print "C"

Picture1.CurrentX = xD: Picture1.CurrentY = yD

Picture1.Print "D"

Picture1.CurrentX = xP: Picture1.CurrentY = yP

Picture1.Print "P"

Picture1.CurrentX = xQ: Picture1.CurrentY = yQ

Picture1.Print "Q"

Picture1.CurrentX = xR: Picture1.CurrentY = yR

Picture1.Print "R"

For wt = 0 To wtt: Next

Next

End Sub

Private Sub Command2_Click()

End

End Sub

Private Function shishagonyuu(ByVal x As Double) As Double

Dim keta As Long: keta = 1000000

shishagonyuu = Int(x * keta + 0.5) / keta

End Function

Private Function inf(ByVal x As Double) As Double

If Abs(x) > 2 Then inf = Sgn(x) * 2 Else inf = x

End Function

問題４プログラム

Option Explicit

Const pi = 3.14159265359

Const r = 1

Sub Form_Load()

Dim waku As Double: waku = 1.5 * r

Picture1.Scale (-waku, waku)-(waku, -waku)

End Sub

Sub Command1_Click()

'm26_4

'　辺ＤＦ＝８，辺ＥＧ＝２のとき，辺ＡＦの長さを求めてください．

Dim j, iro0, iro1, owari As Integer: iro0 = 50: iro1 = 200

Dim wt, wtt As Long: wtt = 10000000

Dim w, tA, tB, tC, tD, tE, kizami As Double

w = pi / 180: kizami = 0.1

Dim xA, yA, xB, yB, xC, yC, xD, yD, xE, yE, xF, yF, xG, yG As Double

Dim range(2), sa(2), DF, GE, AF, a, b, gosa As Double: gosa = 10 ^ (-6)

For tD = 180 - kizami To 90 + kizami Step -kizami

tE = 180 - tD

xD = r * Cos(tD * w): yD = r * Sin(tD * w)

xE = r * Cos(tE * w): yE = r * Sin(tE * w)

range(0) = tE + kizami: range(1) = tD - kizami: owari = 0

While owari = 0: range(2) = (range(0) + range(1)) * 0.5

For j = 0 To 2: tA = range(j)

tB = 2 * tD - tA: tC = 2 * tE - tA

xA = r * Cos(tA * w): yA = r * Sin(tA * w)

xB = r * Cos(tB * w): yB = r * Sin(tB * w)

xC = r * Cos(tC * w): yC = r * Sin(tC * w)

a = (yB - yA) / (xB - xA): b = yA - xA * a

xF = (yD - b) / a: yF = yD

a = (yC - yA) / (xC - xA): b = yA - xA * a

xG = (yE - b) / a: yG = yE

DF = xF - xD: GE = xE - xG

sa(j) = DF / GE - 8 / 2

Next

If Abs(sa(2)) < gosa Or range(1) - range(0) < gosa Then

owari = 1

ElseIf Sgn(sa(2) * sa(0)) < 0 Then

range(1) = range(2)

Else

range(0) = range(2)

End If

Wend

If Abs(sa(2)) < 1 Then

AF = Sqr((xF - xA) * (xF - xA) + (yF - yA) * (yF - yA))

Picture2.Cls: Picture2.Print "DF/GE ="; shishagonyuu(DF / GE)

Picture3.Cls: Picture3.Print "AF="; shishagonyuu(8 * AF / DF); "�p"

Picture4.Cls: Picture4.Print "半径="; shishagonyuu(8 * r / DF); "�p"

Picture1.Cls: Picture1.Circle (0, 0), r, iro0

Picture1.Line (xB, yB)-(xA, yA), iro1

Picture1.Line -(xC, yC), iro1

Picture1.Line (xD, yD)-(xE, yE), iro1

Picture1.CurrentX = xA: Picture1.CurrentY = yA

Picture1.Print "A"

Picture1.CurrentX = xB: Picture1.CurrentY = yB

Picture1.Print "B"

Picture1.CurrentX = xC: Picture1.CurrentY = yC

Picture1.Print "C"

Picture1.CurrentX = xD: Picture1.CurrentY = yD

Picture1.Print "D"

Picture1.CurrentX = xE: Picture1.CurrentY = yE

Picture1.Print "E"

Picture1.CurrentX = xF: Picture1.CurrentY = yF

Picture1.Print "F"

Picture1.CurrentX = xG: Picture1.CurrentY = yG

Picture1.Print "G"

For j = 0 To wt: Next

End If

Next

End Sub

Private Sub Command2_Click()

End

End Sub

Private Function shishagonyuu(ByVal x As Double) As Double

Dim keta As Long: keta = 100000

shishagonyuu = Int(x * keta + 0.5) / keta

End Function

＜水の流れ：コメント＞　８月２４日

浜田さんから、パソコンのプログラム（ＶＩＳＵＡＬ　ＢＡＳＩＣ）を頂いていますが、

不勉強をお許しください。動画画像が皆さんに見てもらえないのが、心苦しいです。