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Question
Determine whether the sets of points are collinear?
(a, b + c), (b, c + a) and (c, a + b)
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Solution
Let the points be A(a, b + c), B(b, c + a) and C(c, a + b)
Area of the triangle = `1/2[(x_1y_2 + x_2y_3 + x_3y_1) - (x_2y_1 + x_3y_2 + x_1y_3)]`
= `1/2["a"("c" + "a") + "b"("a" + "b") + "c"("b" + "c") - "b"("b" + "c") + "c"("c" + "a") + "a"("a" + "b")]`
= `1/2["ac" + "a"^2 + "ab" + "b"^2 + "bc" + "c"^2 - ("b"^2 + "bc" + "c"^2 + "ac" + "a"^2 + "ab")]`
= `1/2 xx 0`
= 0
Since the area of a triangle is 0.
∴ The given points are collinear.
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