Adding a form module to a specific page

Ok, so I need to add a custom form to my user profile page in Drupal. I.e. the /user/1/edit/ form.
I don't want to just alter the existing form because I need it to have a different submission actions.
So I'm going to create a custom module for the form. This is an example I have found on the internet:

<?php

function example_help($section) {
  switch ($section) {
    case 'admin/modules#description':
      return t('This module implements an example form.');
  }
}

function example_menu($may_cache) {
  $items = array();
  if ($may_cache) {
    $items[] = array(
      'path' => 'example',
      'title' => t('Example'),
      'callback' => 'example_page',
      'access' => TRUE,
      'type' => MENU_CALLBACK
    );
  }
  return $items;
}

function example_page() {
  return drupal_get_form('example_page_form');
}

function example_page_form() {
  $form['fullname'] = array(
    '#type' => 'textfield',
    '#title' => t('Enter your full name'),
  );
  $form['submit'] = array(
    '#type' => 'submit',
    '#value' => t('Save'),
  );
  return $form;
}

function example_page_form_submit($form_id, $form_values) {
  $message = 'You have submitted the ' . $form_id . ' form which contains the following data:<pre>' . print_r($form_values,true) . '</pre>';
  drupal_set_message(t($message));
}

?>

Select allOpen in new window

This is all well and good, but this just adds the form to the menu, and puts it on its own page. How could I get this form inserted into a particular area on my User Profile page?
I don't want to use the webforms module or anything, just want a custom form module inserted on my page.
Thanks,