Space geometric vector It is known that the radius of the circumscribed sphere o of the triangular pyramid P-A B C is 1 and satisfies the vector OA + ob + OC = 0 Then the volume of the regular triangular pyramid P-A B C?

Space geometric vector It is known that the radius of the circumscribed sphere o of the triangular pyramid P-A B C is 1 and satisfies the vector OA + ob + OC = 0 Then the volume of the regular triangular pyramid P-A B C?

OA + ob + OC = 0 indicates that the triangle ABC is on the big circle passing through the center O and is an equilateral triangle;
(why? You can go to the midpoint D of AB, then the 2od vector = - OC vector) to show that OCD is collinear and C and D are on the plane ABC, then o will also be on ABC, then | OA | = | ob | = | OC|
Next, use OA + ob + OC = 0 to shift the square of both sides of the term, and you can get that the angle between the two is 120 degrees
It's easy in the back!
Then the volume is good: v = 1 / 3 * [(√ 3 / 4) * (√ 3) ^ 2] * 1 = √ 3 / 4