I get 404 when loading the main page with Index.php, but index.php works with direct links, why?

I can’t understand why mine Homepagesuddenly refused to load ie: directly www.mysite.com/, but it loads only if you typewww.mysite.com/index.php

This is the site:

COMPARISON

http://www.propasiarealty.com // Sends you to the 404 page

BECAUSE THE

http://www.propasiarealty.com/index.php // Loads the homepage.

It is impossible to find out what exactly is wrong, please help.

Below is a screenshot of Filezilla:

EDIT 01

Check out my HTACCESS ... Snapshot

UPDATE 02  I have changed HTACCESS so far without success, now it looks like this:

DirectoryIndex index.php

#ErrorDocument 400     /400.html

#ErrorDocument 401     /401.html 

ErrorDocument 403     /errs/404.php

ErrorDocument 404     /errs/404.php

#ErrorDocument 500     /500.html

Options -Indexes

But still not working

UPDATE 03 [It worked]

The solution suggested by @HPierce was to modify HTACCESS to look something like this:

DirectoryIndex index.php


RewriteEngine on  

RewriteRule ^$ index.php

    #ErrorDocument 400     /400.html

    #ErrorDocument 401     /401.html 

    ErrorDocument 403     /errs/404.php

    ErrorDocument 404     /errs/404.php

    #ErrorDocument 500     /500.html

    Options -Indexes