Albime triangles and Guy’s favourite elliptic curve

This text discusses triangles with the property that a bisector at one vertex, the median at another, and the altitude at the third vertex are concurrent. It turns out that since the 1930s, such triangles appeared in the problem sections of various journals. We recall their well known relation with points on a certain elliptic curve, and we present an elementary proof of the classical result providing the group structure on the real points of such an elliptic curve. A consequence of this answers a question posed by John P. Hoyt in 1991.