  Theorem. There are general position points A, B, C, P on the projective
plane. Let A_P be the intersection point of lines AP and BC. Analogously define
B_P and C_P. Take any points A_1, B_1, C_1 on AP, BP, CP, respectively. Let W_C
be the intersection point of A_1B_P and B_1A_P. Analogously define points W_A
and W_B. Then lines CW_C, AW_A and BW_B pass through one point.
  We also generalize this theorem and find interesting related properties.
