For three non-collinear points

MATHEMATICS · Subjective · CBSE · Mathematics · Coordinate Geometry · Coordinate Geometry

Problem

For three non-collinear points A(x1,y1)A(x_1,y_1), B(x2,y2)B(x_2,y_2) and P(x,y)P(x,y) form the triangle △ABP\triangle ABP. (i) Write the algebraic expression for 2 Area(△ABP)2\,\text{Area}(\triangle ABP) in terms of the coordinates x1,y1,x2,y2,x,yx_1,y_1,x_2,y_2,x,y. (ii) Explain how the sign of this expression changes when the order of the vertices is reversed. (iii) Using the result of part (i), show logically that Area(△ABP)=0\text{Area}(\triangle ABP)=0 if and only if the three points are collinear.

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