Answer to Question #182904 in Analytic Geometry for Keep smile

Question #182904

If A,B,C,D,P,Q are six distinct collinear points ,show that

(AP.AQ)/(AB.AC.AD)+(BP.BQ)/(BC.BD.BA)+(CP.CQ)/(CD.CA.CB)+(DP.DQ)/(DA.DB.DC)=0


1
Expert's answer
2021-05-04T12:06:01-0400

Solution

Since collinear points are points that lie on a straight line, the 6 distinct points A, B, C, D, P, Q lie on a straight line with an equal interval.

Let the interval be x, such that;


"\\frac{(4x) \\cdot(5x)}{(x) \\cdot (2x )\\cdot (3x)}+\\frac{(3x) \\cdot (4x)}{(x) \\cdot (2x)\\cdot (-x)}+\\frac{(2x )\\cdot (3x)}{(x) \\cdot (-2x) \\cdot (-x)}+\\frac{(x) \\cdot (2x)}{(-3x) \\cdot (-2x) \\cdot (-x)}=0\\\\\n\n\\frac{20x^2}{6x^3}+\\frac{12x^2}{-2x^3}+\\frac{6x^2}{2x^3}+\\frac{2x^2}{-6x^3}=0\\\\\n\\frac{20x^2}{6x^3}-\\frac{2x^2}{6x^3}+\\frac{6x^2}{2x^3}-\\frac{12x^2}{2x^3}=0\\\\\n\\frac{18x^2}{6x^3}-\\frac{6x^2}{2x^3}=0\\\\"

Hence Proven


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