child widget does not display in parent widget

I create a parent widget and then a child widget. In the child's paintEvent() method, I simply draw a rectangle. I show() the child widgent and I see it's paintEvent() method called, but I do not see the rectangle drawn. I'm sure I missed something simple, but just don't know what. Thanks for your help.

Code:

// main.cpp
 
#include <QApplication>
#include "ChildTest.h"
 
int main(int argc, char** argv)
{
    // start the display application
    QApplication app(argc, argv);
    ChildTest  child;
 
    child.show();
 
    return app.exec();;
}
 
// ChildTest.h
 
#ifndef CHILDTEST_H
#define CHILDTEST_H
 
#include "ChildPart.h"
 
class ChildTest
{
public:
    ChildTest();
    void show();
 
private:
    QWidget*    m_mainWindow;
    ChildPart*  m_child;
};
 
#endif // CHILDTEST_H
 
// ChildTest.cpp
 
#include "ChildTest.h"
 
ChildTest::ChildTest()
{
    m_mainWindow = new QWidget();
    m_mainWindow->resize(800, 400);
    m_mainWindow->show();
 
    m_child = new ChildPart(m_mainWindow);
}
 
void ChildTest::show()
{
    m_child->show();
}
 
// ChildPart.h
 
#ifndef CHILDPART_H
#define CHILDPART_H
 
#include <QWidget>
 
class ChildPart : public QWidget
{
    Q_OBJECT
 
public:
    ChildPart(QWidget* parent = 0);
 
 
protected:
    void paintEvent(QPaintEvent* event);
 
private:
    QRect   m_rect;
};
 
#endif // CHILDPART_H
 
// ChildPart.cpp
 
#include <QtGui>
#include "ChildPart.h"
#include <QDebug>
 
ChildPart::ChildPart(QWidget *parent) :
    QWidget(parent)
{
}
 
void ChildPart::paintEvent(QPaintEvent*)
{
    qDebug() << "ChildPart::paintEvent  ";
 
    QPainter painter(this); 
    painter.setBrush(Qt::green);
    painter.drawRect(QRect(100, 100, 50, 50));
}