In △ ABC, a, B and C are the opposite sides of angles a, B and C respectively. Given vector M = (a, b), vector n = (COSA, CoSb), vector p = (2 √ 2Sin (B + C) / 2,2sina), if vector m ‖ vector n, vector p ^ 2 = 9, we prove that △ ABC is an equilateral triangle

In △ ABC, a, B and C are the opposite sides of angles a, B and C respectively. Given vector M = (a, b), vector n = (COSA, CoSb), vector p = (2 √ 2Sin (B + C) / 2,2sina), if vector m ‖ vector n, vector p ^ 2 = 9, we prove that △ ABC is an equilateral triangle

OB = (2,0) indicates that the coordinate of point B is (2,0)
OC = (2,2) indicates that the coordinate of point C is (2,2)
CA = (root 2 · cos α, root 2 · sin α), which means that point a is on a circle with the center of point C and the radius of root 2. Let this circle be circle C
The angle between OA and ob is the angle between OA and x-axis
Let the root sign be sqrt ()
Make the line od tangent to the circle C near point B, the tangent point is D, connect CD, then OC = 2sqrt (2) CD = sqrt (2), then sin angle cod = 1 / 2, then angle cod = 30 degrees
Similarly, if the line OE is tangent to the circle C far away from point B, and the tangent point is e, then OC = 2sqrt (2) ce = sqrt (2), then sin angle COE = 1 / 2, then the angle COE = 30 degrees
When the angle cob is 45 degrees, the foot DOB is 15 degrees
Then the range is {15 degrees, 75 degrees]