1. vector A 2 is point a times a, that is, the modulus of a times the modulus of a and then times the cosine of the included angle. Since the included angle is 0 degrees, A 2 = the square of the modulus.

2.( 1)O is the epicenter (the center of the circumscribed circle, the intersection of the vertical lines), because OA=OB=OC. For example, if you cross O as the vertical line of AB, because OA=OB, this vertical line must also bisect AB (the nature of isosceles triangle). Similarly, if we do similar work for AC and BC, it can be proved that O is the intersection of the perpendicular lines of three sides.

(2)N is the center of gravity (the intersection of center lines). According to the parallelogram rule of vector summation, the sum of vector NA+ vector NB (negative vector NC) is the diagonal of the parallelogram on the opposite side composed of NA and NB, and the other diagonal of this parallelogram is AB. Diagonal lines of parallelogram score each other, so we can know that NC crosses the midpoint of AB. Similarly, we can know the situation of other sides, and we can know that n is the intersection of the median lines of three sides.

(3)P is the vertical center (vertical intersection), P * PB = PB * PC means PB*(PA-PC)=0, and PA-PC is the vector AC. The inner product of PB and AC is 0, which means that Pb is perpendicular to AC, so Pb is vertical. Similarly, it can be seen that p is the intersection of three perpendicular lines.