The basic theorem of * * line vector is: If a≠0, then the necessary and sufficient condition of vector B and a*** line is that there is a unique real number λ, so that B = λ a..
The proof process is as follows:
Let a, b and c three-point * * line, and o is any point on the plane.
Because lines A, B, C*** have a non-zero real number k, so that
AB = KAC
That is, OB-OA=k(OC-OA)
So OB=kOC+( 1-k)OA.
[Note: the sum of the two coefficients is k+ 1-k= 1]
On the other hand, if there is a real number x, y satisfies x+y= 1 and OA=xOB+yOC.
Then OA=xOB+( 1-x)OC.
OA-OC=x(OB-OC)
So CA=xCB
Therefore, the lines of vectors CA and CB * * *
BeCAuse ca and CB have one thing in common, C.
So, the three-point line of a, b and c.
Proof method of three-point * * * line:
Method 1: take two points to establish a straight line and calculate the analytical formula of the straight line? Substitute the coordinates of the third point to see if the analytical formula (straight line and equation) is satisfied.
Method 2: Let three points be A, B and C, and prove by vector: λAB=AC (where λ is a non-zero real number).
Method 3: Calculate AB slope and AC slope by the point difference method, which is equal to the three-point * * * line.
Method 4: Using Menelaus Theorem.
Method 5: Using the axiom in geometry that "if two non-overlapping planes have a common point, they have only one common straight line passing through the point", we can know that if three points belong to two intersecting planes, they are a straight line.
Method 6: Apply the axiom that there is only one straight line parallel (perpendicular) to the known straight line at a point outside the straight line. Actually, it's the same method.
Method 7: Prove that the included angle is 180.
Method 8: Let A B C and prove that the area of △ABC is 0.