Question

M ìs the midpoint of line segment AB, AXB and MYB are equilateral triangles on opposite sides of AB, XY cuts AB at Z
Prove that - AZ= 2 ZB
PROVE BY SIMILARITY.
PLZ reply fast...
It's urgent

Answer 1

Since M is the mid point of AB , AM = MB
AB = 2 AM = 2 MB

Since ABX and ABY are equilateral , AB = BX = AX = 2MY = 2BY

Triangles BXZ and BZY have congruent vertical angles at Z , and congruent 60° angles ZBX and ZMY .
they are similar , and since BX = 2BY ,ZB = 2MZ
then AM = MB = MZ +ZB = 3 MZ
so, AZ = AM +MZ = 3MZ+MZ = 4MZ ,
and since ZB = 2MZ ,AZ/ZB = 2MZ / 2MZ = 2
AZ / ZB = 2
AZ = 2ZB