In \triangle ABC, segment CD bisects \angle C and intersects AB at point D. Show that CD^2 < CA\cdotp CB

Show the following inequality

1-\dfrac{1}{6n^2}-\dfrac{1}{6n^3} <\displaystyle\sum_{k=1}^n \sqrt[3]{ n^3+k} <1-\dfrac{1}{6n^2}

If the greatest common divisor (GCD) of a , b is 1, c, Verify a-b is a perfect square such that \dfrac{ab}{a-b} =c

Find the range of the function y = \sin x+ \cos x+\sin x\cos x

What are Types of Polygons

Find the coefficient of x^{49} in the polynomial p(x)=(x–1)(x–2)(x–3) \dots(x-50)

The perimeter of diamond ABCD is 16 cm,

∠ABC＝60° , the diagonal lines AC and BD intersects at point O .

Find the length of AC and BD

Given the right trapezoid ABCD , AB\parallel CD , AB\perp AD , AB=AD=\dfrac{1}{2}CD=1 . Now make a square ADEF with AD as one of its sides, and then fold the square along AD to make the plane ADEF is perpendicular to plane ABCD , M is the midpoint of ED .

(1) Prove AM\parallel plane BEC ;

(2) Prove BC\perp plane BDE ;

(3) Find the distance from D to plane BEC

Given the base of pyramid P-ABCD is a square. If the side of the square is 1 in length , PA\perp CD , PA=1 , PD=\sqrt{2} .

Prove PA\perp plabe ABCD

In the pyramid P-ABCD , △PBC is a right triangle, AB⊥ plane PBC , AB \parallel CD , AB = \dfrac{1}{2} DC . E is the midpoint of PD . Prove:

(1) AE \parallel plane PBC .

(2) AE ⊥ plane PDC