Question 1154271
<pre>

{{{drawing(200,200,-1,9,-1,9,

line(0,0,8,0),line(8,0,8,8),line(8,8,0,8),line(0,8,0,0),
line(0,0,8,8),line(4,0,0,8), 
locate(0,8.8,A),locate(0,0,B),locate(8,0,C),locate(8,8.8,D),locate(4,0,M),locate(3,3,N) )}}}

Draw PQ through N parallel to AB

{{{drawing(200,200,-1,9,-1,9,
green(line(8/3,0,8/3,8),locate(8/3,8.8,P),locate(8/3,0,Q)),
line(0,0,8,0),line(8,0,8,8),line(8,8,0,8),line(0,8,0,0),
line(0,0,8,8),line(4,0,0,8), 
locate(0,8.8,A),locate(0,0,B),locate(8,0,C),locate(8,8.8,D),locate(4,0,M),locate(3,3,N) )}}} 

{{{AP/(MQ)=PD/(QB)}}}

{{{AD=AP+PD=8}}}     {{{MB=MQ+QB=4}}}
   {{{PD=8-AP}}}        {{{QB=4-MQ}}}

{{{AP/(MQ)=(8-AP)/(4-MQ)}}}

{{{AP(4-MQ)=MQ(8-AP)}}}

{{{4*AP-AP*MQ=8*MQ-MQ*AP}}}

{{{4*AP-AP*MQ=8*MQ-MQ*AP}}}

{{{4*AP-AP*MQ=8*MQ-AP*MQ}}}

{{{4*AP=8*MQ}}}

{{{AP=2*MQ}}}

By similar triangles,

{{{NA=2*NM}}}

By the Pythagorean theorem,

{{{AM^2=AB^2+BM^2}}}
{{{AM^2=8^2+4^2}}}
{{{AM^2=64+16}}}
{{{AM^2=80}}}
{{{AM=sqrt(80)}}}
{{{AM=sqrt(16*5)}}}
{{{AM=4sqrt(5)}}}
{{{NA+NM=4sqrt(5)}}}
{{{2*NM+NM=4sqrt(5)}}}
{{{3*NM=4sqrt(5)}}}
{{{NM=4sqrt(5)/3}}}
{{{NA=2*NM=2*4sqrt(5)/3=8sqrt(5)/3}}}

You finish by finding NB and ND the same way.

Edwin</pre>