Orthocenter Symmetry

AKA- Johnson circles, or The 5 lei coin problem

The goal of this page is to prove the property found on the page above (https://www.geogebra.org/m/A2NGCM3K). Use the scroll bar to unfold the idea behind the proof (or rather, one of the proofs. The law of sines provides a much shorter proof).
pause. why does ? once you have answered that question you may continue: