Answer to Question #182904 in Analytic Geometry for Keep smile

Question

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

Answer 1

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;

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

Hence Proven