"I don't get why we teach special features of triangles like centroids, circumcenters and incenters, and the Hinge Theorem, when they seem like pretty useless things to know that don't even impart any meaningful thinking/reasoning skills."
Project for Incenter: http://goo.gl/j3SqTK (think efficient packing)
also good cross reference for congruent triangles and/or kite symmetry.
Project for Circumcenter: http://goo.gl/2E9faE
Centroid gives balance point with equal areas. Can have participants cutout triangles of varying size. Find centroid and hang from point like a mobile.