Let P be a point. The parallel to BC from P cuts AB at Ba and AC at Ca. Define cyclically Cb, Ab, Ac, Bc. We have the following relations on segment lengths:
if P = X(75), AbBa = BcCb = CaAc
if P = K (symmedian), AcAb = BaBc = CbCa
if P = Mt (Mittenpunkt), AAc+AAb = BBa+BBc = CCb+CCa
if P = Sp` (isotomic of the Spieker point Sp), AAc+BBc = BBa+CCa = CCb+AAb