Let △ABC be a triangle in the projective plane P² . On line BC is a point D (unequal to B or C), on the line CA is a point E (unequal to C or A) and on the line AB is a point F (unequal to A or B). Write P = BC ∩ EF, Q = CA ∩ F D and R = AB ∩ DE. If the lines AD, BE and CF pass through a point, then P, Q and R are collinear. To prove this statement, we assume that the lines AD, BE and CF pass through go one point; we call this point S. Then A, B, C, S is a projective frame for P². We will work in this exercise relative to this framework, i.e., we will assume that: A = (1 : 0 : 0), B = (
In this exercise we prove the following theorem (see also the attached image):
Let △ABC be a triangle in the projective plane P² . On line BC
is a point D (unequal to B or C), on the line CA is a point E (unequal to C or A) and on the line AB is a point F (unequal to
A or B). Write P = BC ∩ EF, Q = CA ∩ F D and R = AB ∩ DE.
If the lines AD, BE and CF pass through a point, then P, Q and R are collinear.
To prove this statement, we assume that the lines AD, BE and CF pass through
go one point; we call this point S. Then A, B, C, S is a projective frame
for P². We will work in this exercise relative to this framework, i.e., we will assume that:
A = (1 : 0 : 0), B = (0 : 1 : 0), C = (0 : 0 : 1) and S = (1 : 1 : 1).
Determine the coordinates of points P, Q and R.
if able please provide some explanation with the taken steps, thank you very much.
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