This is the top of a tilt-top-table. Because the broken piece holds the entire weight of the oak table top it had to be replaced instead of repaired. The trick with such a repair is to make it as strong or stronger than the original and also to match it in to disguise the repair to keep the aesthetic appeal.
This involved cutting and planing flush the original piece and making a new piece in approximately the correct dimensions. Then dowelling and gluing this together to give extra strength. Then carving the profile of the original piece onto the new. Including carving two round spigots to allow the top to tilt. After this the alchemy can begin of matching in the colours of the old and the new wood. The finished product is shown on the last two photos.