Midpoint of two line segments in three dimensions

Question

This might be an easy question, but since i'm new to solid shapes, i couldn't solve it. A= (7,1,3) B=(5,1,2) C=(4,-2,3) D=(6,m,n) I need to find m and n so that segments BD and AC have the smae midpoint. I wrote the midpoint formula for AC and found its midpoints and then tried to equate it with the midpoints of the segment DC, but i have 2 unknowns there and couldn't find m and n. Any help would be appreciated

Answer 1

The midpoint of $AC$ has coordinates: $$ M_{AC}=\left(\frac{x_A+x_C}{2},\frac{y_A+y_C}{2},\frac{z_A+z_C}{2}\right)=\left(\frac{11}{2},-\frac{1}{2},3 \right) $$ In the same way,the midpoint of $BD$ has coordinates: $$ M_{CD}=\left( \frac{11}{2},\frac{1+m}{2},\frac{2+n}{2}\right) $$

so we want: $$ \frac{1+m}{2}=-\frac{1}{2} \quad \land \quad \frac{2+n}{2}=3 $$ two equations for two unknowns.