If z1 = x1 + iy1, z2 = x2 + iy2, then distance between points z1, z2 is argard plane is |z1-z2|= √((x1-x2)2 + (y1-y2)2)
In triangle OAC,
OC ≤ OA + AC
OA ≤ AC + OC
AC ≤ OA + OC
Using these inequalities we have
||z1| - |z2|| ≤ |z1+z2| ≤ |z1| + |z2|
Similarly from triangle OAB, we have
||z1| - |z2|| ≤ |z1-z2| ≤ |z1| + |z2|
Note:
(a) ||z1| - |z2|| = |z1+z2| , |z1-z2| = |z1| + |z2| iff origin, z1, and z2 are collinear and origin lies between z1 and z2.
(b) |z1 + z2| = |z1|+|z2| , ||z1| - |z2|| = |z1-z2| iff origin, z1 and z2 are collinear and z1 and z2 lies on the same side of origin.
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