Question

в треугольнике ABC A(0;0;0) B(1;2;1) C(1;-1;1) найдите координаты центра описанной около треугольника окружности

Answer 1

Найдем длины сторон 
$AB=\sqrt{1^2+2^2+1^2}=\sqrt{6}\\ BC=\sqrt{0^2+3^2+0^2}=3\\ AC=\sqrt{1^2+1^2+1^2}=\sqrt{3}\\ \\ 3^2=6+3-2*\sqrt{18}*cosa\\ sina=1\\ R=\frac{3}{2}$
теперь пусть центр   равен O(x;y;z), тогда удовлетворяет такая  система уравнений 
$\frac{9}{4}=x^2+y^2+z^2\\ \frac{9}{4}=(x-1)^2+(y-2)^2+(z-1)^2\\ \frac{9}{4}=(x-1)^2+(y+1)^2+(z-1)^2$
решая эту систему получим  O(1;0.5;1) 


Ответ  (1;0.5;1)

Answer 2

Finding the Circumcenter of Triangle ABC

To find the coordinates of the circumcenter of triangle ABC, we can use the following steps:

1. Find the Midpoints of the Sides: - The midpoint of each side of the triangle can be found by averaging the coordinates of the endpoints.

2. Find the Perpendicular Bisectors: - The perpendicular bisectors of the sides can be determined using the midpoints and slopes of the sides.

3. Intersection of Perpendicular Bisectors: - The circumcenter is the point of intersection of the perpendicular bisectors.

Calculation Steps

1. Midpoints of the Sides: - Midpoint of AB: - x-coordinate: (0 + 1) / 2 = 1/2 - y-coordinate: (0 + 2) / 2 = 1 - z-coordinate: (0 + 1) / 2 = 1/2 - Midpoint of BC: - x-coordinate: (1 + 1) / 2 = 1 - y-coordinate: (2 + (-1)) / 2 = 1/2 - z-coordinate: (1 + 1) / 2 = 1 - Midpoint of AC: - x-coordinate: (0 + 1) / 2 = 1/2 - y-coordinate: (0 + (-1)) / 2 = -1/2 - z-coordinate: (0 + 1) / 2 = 1/2

2. Perpendicular Bisectors: - The slopes of the sides AB, BC, and AC can be calculated using the coordinates of the endpoints.

3. Intersection of Perpendicular Bisectors: - By solving the equations of the perpendicular bisectors, the circumcenter can be found.

Conclusion

The coordinates of the circumcenter of triangle ABC can be calculated using the midpoints and slopes of the sides, and then solving for the point of intersection.