Vector intersection
By Anonymous on Tuesday, February 27,
2001 - 09:31 pm :
A,B,&C are non colinear points in a plane.
C' is mid pt of AB
A' is mid pt of BC
B' is mid pt of CA
Show that the lines passing through A & A', B & B',
C & C' all meet at a point.
By Kerwin Hui (Kwkh2) on Tuesday,
February 27, 2001 - 09:47 pm :
Let O be a fixed point, and let a, b,
c denote the vectors O A, O B, O C respectively. Then,
vector
so equation of the line A A ' is
r1=a+l1(b+c-2a) = (1-2l1)a+l1b+l1c
Similar for B B ' , C C ' and check that l1=l2=l3=1/3
gives the required point.
Alternatively we can get apurely geometric proof as follows: Join the lines
A B, B C, C A, A A ' , B B ' , A ' B ' and let G be the intersection of
A A ' and B B ' . Then, by elementary geometry, we see that
A G/G A ' =BG/B G ' =2, hence result.
Kerwin