mathsen

For which real values of b do the equations x^3 + b x^2 + 2bx - 1 = 0 and x^2 + (b-1)x + b = o have a common root?

In a pentagon ABCDE, AB = AE, BC = ED and BD = EC. Prove that AC = AD.

In a quadrilateral ABCD, the diagonal AC bisects angle BAD and AB = BC = CD. Prove that the other diagonal, BD, bisects angle ADC.

Let P and Q be the points on the sides AB and BC of a triangle ABC respectively such that BP = 3PA and QC = 2BQ. Let K be the midpoint of the segment PQ. Prove that the area of the triangle AKC is equal to 11S/24, where S is the area of the triangle ABC.

Let ABC be a triangle such that angle ACB = 135 degrees. Prove that AB^2 = AC^2 + BC^2 + ¡Ì2 x AC x BC.