More variations on Nagel and Gergonne analogues of the Steiner-Lehmus theorem

The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal then the corresponding sides have equal lengths. That is to say if P is the incentre of Δ ABC and if BP and CP meet the sides AC and AB at B ′ and C ′, respectively, then An elegant proof of this theorem appeared in [1] and is reproduced in [2].