Proof (method 2: vectors viewpoint )

In[]:=
(*definevectors*)​​{a,b,c,d,m,n}={{ax,ay},{bx,by},{cx,cy},{dx,dy},{mx,my},{nx,ny}};​​{ab,an,dn,dc,mb,mc}={a-b,a-n,d-n,d-c,m-b,m-c};
(*solveequations*)​​sol=First@Solve[{(mx-ax)/(dx-mx)==(my-ay)/(dy-my)==(nx-cx)/(bx-nx)==(ny-cy)/(by-ny)λ,​​(dx-mx)≠0,(dy-my)≠0,(bx-nx)≠0,(by-ny)≠0},{nx,ny,mx,my}]
Out[]=
nx-
-cx-bxλ
1+λ
,ny-
-cy-byλ
1+λ
,mx-
-ax-dxλ
1+λ
,my-
-ay-dyλ
1+λ

(*determinationcalculatearea:Δabn+Δdcn*)​​area1=0.5(Det[Transpose[{ab,an}]]+Det[Transpose[{dn,dc}]])
Out[]=
0.5(-aybx+axby-cydx+cxdy+aynx-bynx+cynx-dynx-axny+bxny-cxny+dxny)
(*determinationcalculatearea:Δbmc*)​​area2=0.5Det[Transpose[{mb,mc}]]
Out[]=
0.5(-bycx+bxcy+bymx-cymx-bxmy+cxmy)
In[]:=
Simplify[(area1-area2)/.sol]
Out[]=
0.
In[]:=
Simplify[(area1/area2)/.sol]
Out[]=
1.