Given four concyclic points. For each subset of three points take the incenter. Show that the four incentres form a rectangle.

Can a pentagon with all rational side lengths be inscribed in a circle of radius ?

Let and Prove that
Where

Let be a trapezoid such that lie on a circle with center , and side is parallel to side . Diagonals and intersect at point , and . It is given that . It is also known that the difference in length between and can be expressed in the form , where and are positive integers and is square-free. Compute the value of .

Let Prove that

1. Let r be the radius of the largest circle which is tangent to the parabola y=x^2 at x=0 and which lies entirely on or inside (that is, above) the parabola, find r.

2. Counting number n has the following property,: if we take any 50 different numbers from 1,2,3, ... n, there always are two numbers with the difference of 7. what is the largest value among all value of n?


thanks a lot

IMAGE

The points in this are given by
P = (sin2A, sin4A)∀A [0,2π]
Is this a conic section? what is this?

A sphere is inscribed inside a cone with base radius and height Another sphere of radius is internally tangent to the lateral surface of the cone, but does not intersect the larger inscribed sphere. A plane is tangent to both of these spheres, and passes through the inside of the cone. The intersection of the plane and the cone forms an ellipse. Find the maximum area of this ellipse.

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Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!