Place the medallion on the floor where you intend to install it, and trace a line around the perimeter of the medallion onto the floor with a metal scribe. The scribe leaves an impression in the surface of the wood.
Secure an inlay router bit into your router and set the depth of the router bit to match the thickness of the medallion. Since the medallion is made of tile, you cannot sand it after installation. Therefore, the depth hole you route for inlaying the medallion must match the thickness of the medallion.
Route the hole for the medallion with the router. Cut as close to the inside of the line as possible to create the tightest fit that you can.
Apply a small amount of epoxy into the hole you routed so you have a thin layer covering the floor of the hole.
Press the medallion inlay into the hole and quickly wipe away any excess epoxy that seeps out with a moist rag. Allow the epoxy to cure for 24 hours.
Fill in any gaps between the sidewall of the hole and the edges of the medallion with inlay filler. Level the filler with the surface of the floor and medallion with your finger and allow an additional 24 hours of drying time.