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Question
Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.
Solution
It is known that the vertices of the triangles A (a, b + c), B (b, c + a) and C (c, a + b)
`Delta` area of `= Delta = 1/2 abs (("x"_1,"y"_1,1),("x"_2,"y"_2,1),("x"_3,"y"_3,1))`
x1 = a,y1 = b + c, x2 = b, y2 = c + a, x3 = c, y3 = a + b
`= 1/2 abs (("a", "b + c", 1),("b", "c + a",1),("c", "a + b", 1)) ...("C"_1 -> "C"_1 + "C"_2)`
`= 1/2 abs (("a + b + c", "b + c", 1),("a + b + c", "c + a", 1),("a + b + c", "a + b", 1))`
`= 1/2 ("a + b + c") abs ((1, "b + c", 1),(1, "c + a", 1),(1, "a + b", 1))`
`= 1/2 ("a + b + c") xx 0 ...("C"_1 "and" "C"_2 "are the same")`
`Delta` area of = 0
Hence, points A, B, C are collinear.
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