Let , and be the vertices of a triangle , and let be any point in the…

MATHEMATICS · Multi-part · CBSE · Mathematics · Coordinate Geometry · Coordinate Geometry

Problem

Let A(x1,y1)A(x_1,y_1), B(x2,y2)B(x_2,y_2) and C(x3,y3)C(x_3,y_3) be the vertices of a triangle △ABC\triangle ABC, and let P(xp,yp)P(x_p,y_p) be any point in the plane. (a) Prove that Area(PBC)+Area(PCA)+Area(PAB)=Area(ABC)\text{Area}(PBC)+\text{Area}(PCA)+\text{Area}(PAB)=\text{Area}(ABC) regardless of the position of PP. (b) Deduce from part (a) that, when PP lies strictly inside △ABC\triangle ABC, each of the ratios Area(PBC)Area(ABC),  Area(PCA)Area(ABC),  Area(PAB)Area(ABC)\frac{\text{Area}(PBC)}{\text{Area}(ABC)},\;\frac{\text{Area}(PCA)}{\text{Area}(ABC)},\;\frac{\text{Area}(PAB)}{\text{Area}(ABC)} is positive and that their sum equals 11. (c) If PP lies on the side BCBC, identify which of the three ratios in part (b) becomes zero and verify that the sum of the ratios remains 11.

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