Line is perpendicular to line and is 90 degrees. We know this because line intercepts an arc of 180 degrees. An inscribed angle is half the measure of the intercepted arc. Therefore this must be a right triangle.

This is demonstrated by the inscribed angle on a circle. An inscribed angle is an angle whose vertex is on the circumference of the circle and whose sides extend as chords of the circle. This theorem is proven to be true and can be used to solve for angles of inscribed figures in circles. This theorem is portrayed in the figure below.

Explanation: The steps are shown below along with figures for more explanation

1. Make a point at any point on the circle’s circumference

2. Draw an arc across the circle with a compass that is set to the length of the radius, this will be the next vertex

3. Draw arcs in this fashion until there are six vertices

4. These are the six vertices of the hexagon

5. Connect these six vertices making three equal sides

Since we are finding the side lengths of a square, we really only need to find one side since all sides are congruent in a square. Recall that the formula for the area of a triangle is . We can find the area of the square by finding the area of the triangle and subtracting the area of the square. Having the area of the square will just be one of the side lengths squared, so if we take the square root of the area of the square, we will have our answer.

First, consider the small triangle in the upper corner. This is shown below:

The area of this triangle would be

We will consider the lower two triangles combined to be the second triangle:

The area of this triangle would be

And then we need to consider a third area which is the square

The area of the square would be

To get the total area of the triangle, we can sum all three areas.

We can sub in our areas and solve for to solve this problem

(multiply by 2 to get rid of fractions)

Since is the radius, we can extend it to represent the diameter which is .

We know that a square is made up of two right triangles. So the diagonal must be a product of the Pythagorean Theorem; . We should let and since we are working with a square we know . We can simplify this formula to be

If we take the square root of each side we get

The area of a square is just . So the area of this square in terms of is

Explanation: The steps are shown below along with figures for more explanation

1. Make a point at any point on the circle’s circumference

2. Draw an arc across the circle with a compass that is set to the length of the radius, this will be the next vertex

3. Draw arcs in this fashion until there are six vertices

4. Label every other vertex so that there are three vertices

5. Connect these three vertices making three equal sides

Explanation: The steps are shown below along with figures for more explanation

1. Given a circle (or draw a circle) draw a line across the diameter

2. Draw a line perpendicular to the diameter that also bisects the diameter

3. Label all points that intersect the circumference of the circle

4. Connect the points on the outer edge of the circle to form the four sides of the square

First understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a rectangle inscribed in a circle means the rectangle will be drawn inside of a circle.

From here the following image can be constructed.

The diameter of the circle is any and all straight lines that cut the circle in half. Since the rectangle is inscribed in that circle and all corners touch the circle then the diameter of the circle is equivalent to the length of the rectangle's diagonal.

This becomes clear when demonstrated through a picture. Not only does this allow us to inscribe a square within a right triangle, but it also gives us the largest square possible within the right triangle.

The incenter is the intersection of the triangle’s three angle bisectors. Drawing two of these angle bisectors is sufficient enough to find the incenter. Below is a figure that illustrates the incenter of a triangle as point A.