theorem Th19: :: MENELAUS:19

for A, B, C, A1, B1, C1, A2, B2, C2 being Point of (TOP-REAL 2)
for lambda, mu, nu being Real st A,B,C is_a_triangle & A1 = ((1 - lambda) * B) + (lambda * C) & B1 = ((1 - mu) * C) + (mu * A) & C1 = ((1 - nu) * A) + (nu * B) & lambda <> 1 & mu <> 1 & nu <> 1 & A,A1,B2,C2 are_collinear & B,B1,A2,C2 are_collinear & C,C1,A2,B2 are_collinear holds
( (((1 - mu) + (lambda * mu)) * ((1 - lambda) + (nu * lambda))) * ((1 - nu) + (mu * nu)) <> 0 & the_area_of_polygon3 (A2,B2,C2) = (((((mu * nu) * lambda) - (((1 - mu) * (1 - nu)) * (1 - lambda))) ^2) / ((((1 - mu) + (lambda * mu)) * ((1 - lambda) + (nu * lambda))) * ((1 - nu) + (mu * nu)))) * (the_area_of_polygon3 (A,B,C)) )