Step-by-step explanation:
Let use the first of the medians. Medians AX.
Median AX is the midpoint of BC so our median is
[tex] \frac{1}{2} (b + c) - a[/tex]
Let use the second median.
Median BY is the midpoint of AC so our median is
[tex] \frac{1}{2} (a + c) - b[/tex]
Let use the final median.
Median CZ is the midpoint of AB so our median is
[tex] \frac{1}{2} (a + b) - c[/tex]
Now add all them up
[tex] \frac{1}{2} (b + c ) - a + \frac{1}{2} (a + c) - b + \frac{1}{2} (a + b) - c[/tex]
[tex] \frac{1}{2} b + \frac{1}{2} c - a + \frac{1}{2} a + \frac{1}{2} c - b + \frac{1}{2}a + \frac{1}{2} b - c[/tex]
Add like terms.
[tex] \frac{1}{2} b + \frac{1}{2} b - b = 0[/tex]
[tex] \frac{1}{2} a + \frac{1}{2} a - a = 0[/tex]
[tex] \frac{1}{2} c + \frac{1}{2} c - c = 0[/tex]
[tex]0 + 0 + 0 = 0[/tex]
So
AX+BY+CZ=0
