1. Draw the equilateral triangle (thick black lines)
2. Draw the angle bisectors of the interior angles (blue lines)
3. Draw the angle bisector of the right angle marked in yellow (red line)
4. The interception of the red angle bisector with the blue angle bisector is the center of one of the three common circles.
5. Construct the length of the radius of the interior circle: Draw a line perpendicular to one side of the triangle through the center of the circle.
6. Draw the circle.
7. The remaining centers of the two other circles are the vertices of an equilateral tringle whose sides are parallel to the sides of the given triangle. Construct the centers of the two other circles.
8. I don't understand what you want to proof. So I'll leave this part for you.