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دائرة تمر بثلاث نقاط: المركز ونصف القطر والمعادلة

أوجد مركز الدائرة المارة بثلاث نقاط ونصف قطرها ومساحتها ومعادلتها بالصيغتين القياسية والعامة.

آخر تحديث أمثلة تم التحقق منها: 4

جرّب
نصف القطر
نصف القطر: 5
الحد الأقصى للمنازل العشرية: 8؛ إلى الأقرب، وعند التعادل بعيدًا عن الصفر
Centre x
1
Centre y
1
Standard form
(x − 1)² + (y − 1)² = 25
General form
x² + y² − 2x − 2y − 23 = 0
القطر
10
محيط الدائرة
31.4159
المساحة
78.5398

The only circle through (−3, 4), (4, 5) and (1, −4) is centred at (1, 1) with radius 5: (x − 1)² + (y − 1)² = 25.

Circle through the three points

r = 5centre (1, 1)P1(−3, 4)P2(4, 5)P3(1, −4)
طريقة الحساب S

  1. Determinant (zero means the points are collinear)

    D=2[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]=−120D = 2[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] = -120

  2. Centre

    h=(x12+y12)(y2−y3)+(x22+y22)(y3−y1)+(x32+y32)(y1−y2)D=1k=(x12+y12)(x3−x2)+(x22+y22)(x1−x3)+(x32+y32)(x2−x1)D=1\begin{gathered} h = \frac{(x_1^2 + y_1^2)(y_2 - y_3) + (x_2^2 + y_2^2)(y_3 - y_1) + (x_3^2 + y_3^2)(y_1 - y_2)}{D} = 1 \\[6pt] k = \frac{(x_1^2 + y_1^2)(x_3 - x_2) + (x_2^2 + y_2^2)(x_1 - x_3) + (x_3^2 + y_3^2)(x_2 - x_1)}{D} = 1 \end{gathered}

    The centre is where the perpendicular bisectors of P₁P₂ and P₂P₃ meet, so it is equally far from all three points.

  3. نصف القطر

    r=(x1−h)2+(y1−k)2=25=5r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} = \sqrt{25} = 5

  4. Equation

    Standard: (x − 1)² + (y − 1)² = 25 · General: x² + y² − 2x − 2y − 23 = 0

حول دائرة تمر بثلاث نقاط: المركز ونصف القطر والمعادلة

Three points that do not lie on one line fix exactly one circle, the circumcircle of the triangle they form. Its centre (h, k) is where the perpendicular bisectors of two chords meet, which a determinant formula gives directly, and the radius is the distance from the centre to any of the points. Coordinates stay exact fractions, so a centre at (1, 4/3) is shown as 4/3, not 1.3333.

The default points (−3, 4), (4, 5) and (1, −4) give centre (1, 1) and radius 5: (x − 1)² + (y − 1)² = 25, or x² + y² − 2x − 2y − 23 = 0. The same construction finds the centre of a round table, pipe or arch from three marks on its edge, and the radius of a road curve from three survey points.

If the determinant is zero the points are collinear and no circle exists. Coordinates carry no unit; the radius and area are in the coordinates' unit and its square.

أمثلة محلولة

(−3, 4), (4, 5), (1, −4)

Point 1: x
-3
Point 1: y
4
Point 2: x
4
Point 2: y
5
Point 3: x
1
Point 3: y
-4
Centre x
1
Centre y
1
نصف القطر
5
Standard form
(x − 1)² + (y − 1)² = 25
General form
x² + y² − 2x − 2y − 23 = 0

مصدر التحقق: ⁨Python 3.8 fractions with the determinant formula; each point is 5 from (1, 1): 4² + 3², 3² + 4², 0² + 5²⁩

Right-angled corner (0, 0), (4, 0), (0, 3)

Point 1: x
0
Point 1: y
0
Point 2: x
4
Point 2: y
0
Point 3: x
0
Point 3: y
3
Centre x
2
Centre y
1.5
نصف القطر
2.5
المساحة
19.634954

مصدر التحقق: ⁨Thales: the hypotenuse (length 5) is a diameter, centre at its midpoint; Python 3.8 math: pi*2.5**2⁩

Centre at the origin (edge case: no shift terms)

Point 1: x
1
Point 1: y
0
Point 2: x
0
Point 2: y
1
Point 3: x
-1
Point 3: y
0
Centre x
0
Centre y
0
نصف القطر
1
Standard form
x² + y² = 1
General form
x² + y² − 1 = 0

مصدر التحقق: ⁨All three points are 1 from the origin⁩

Fractional centre (0, 0), (2, 0), (1, 3)

Point 1: x
0
Point 1: y
0
Point 2: x
2
Point 2: y
0
Point 3: x
1
Point 3: y
3
Centre x
1
Centre y
1.33333333
نصف القطر
1.66666667
Standard form
(x − 1)² + (y − 4/3)² = 25/9

مصدر التحقق: ⁨Python 3.8 fractions: h = 1 by symmetry, 9 − 6k = 1 gives k = 4/3, r² = 1 + 16/9 = 25/9⁩

الأسئلة

How do you find the equation of a circle through three points?

Substitute each point into the general form x² + y² + Dx + Ey + F = 0 and solve the three linear equations for D, E and F. For (−3, 4), (4, 5) and (1, −4) this gives D = −2, E = −2 and F = −23. Completing the square turns that into (x − 1)² + (y − 1)² = 25: centre (1, 1), radius 5.

How do you find the centre of a circle from three points on it?

Construct the perpendicular bisectors of two chords, such as P₁P₂ and P₂P₃; they cross at the centre, because every point on a perpendicular bisector is equally far from both ends of its chord. For a right triangle the centre is the midpoint of the hypotenuse: (0, 0), (4, 0) and (0, 3) give centre (2, 1.5) and radius 2.5.

What is the general form of the equation of a circle?

x² + y² + Dx + Ey + F = 0. The centre is (−D/2, −E/2) and the radius is √(D²/4 + E²/4 − F). For x² + y² − 2x − 2y − 23 = 0 the centre is (1, 1) and the radius √(1 + 1 + 23) = 5. If D²/4 + E²/4 − F is zero the equation describes a single point, and if it is negative, no real circle.

Why is there no circle through three points on a straight line?

A circle meets a straight line at most twice, so it cannot pass through three collinear points. In the formula, x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂), which is twice the triangle's signed area, becomes zero and the centre would need a division by zero. Points that are nearly collinear give a very large radius.

ما مدى دقة «⁨دائرة تمر بثلاث نقاط: المركز ونصف القطر والمعادلة⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 4. مثلًا، يجري التحقق من «⁨(−3, 4), (4, 5), (1, −4)⁩» بالرجوع إلى ⁨Python 3.8 fractions with the determinant formula; each point is 5 from (1, 1): 4² + 3², 3² + 4², 0² + 5²⁩.

ما مصدر هذه الطريقة؟

Weisstein, E. W. “Circumcircle” — MathWorld (circle through three points, determinant form); Weisstein, E. W. “Circle” — MathWorld (standard and general equations).

حول هذه الحاسبة

h=∑(xi2+yi2)(yj−yk)2∑xi(yj−yk),k=∑(xi2+yi2)(xk−xj)2∑xi(yj−yk)(x−h)2+(y−k)2=r2\begin{gathered} h = \frac{\sum (x_i^2 + y_i^2)(y_j - y_k)}{2\sum x_i(y_j - y_k)},\quad k = \frac{\sum (x_i^2 + y_i^2)(x_k - x_j)}{2\sum x_i(y_j - y_k)} \\[6pt] (x - h)^2 + (y - k)^2 = r^2 \end{gathered}

المصادر

  1. Weisstein, E. W. “Circumcircle” — MathWorld (circle through three points, determinant form)
  2. Weisstein, E. W. “Circle” — MathWorld (standard and general equations)

تم التحقق بالرجوع إلى المصادر

تتضمن هذه الحاسبة أمثلة محلولة بإجابات من مصادر مستقلة، وعددها 4. تُشغّل ضمن مجموعة الاختبارات، ويمكنك تشغيلها هنا أيضًا.

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