Drum Roll Please: The Debut of TC's Total Challenge

As you may have read in an earlier comment, I've invited one of MathNotations' most dedicated and talented contributors to go beyond commenting and share some of his creative ideas and insights by being an occasional guest blogger - he has graciously accepted.

For his inaugural offering, tc is challenging you and/or your students to solve a classic calculus problem using non-calculus methods. I have made a few minor edits, but the activity is essentially what tc sent to me.
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I give you tc's Total Challenge I:

One of my math professors in college used to say there were three ways
of tackling any problem: the right way, the wrong way and the Navy way
(correct, but extremely roundabout).

In this exercise, we will look at three ways (not necessarily the ones
named above) of doing the following problem:

Determine the rectangle of maximum area that can be inscribed
in a circle of given radius r.

Let the inscribed rectangle have sides a and b. The diagonal of the rectangle passes through the center of the circle (this can be shown, but you can assume it is true).

(1) Express r in terms of a and b.

(2) Express the area in terms of a and r.

(3) Instead of maximizing the area, we can maximize the square of the area.
(a) Express the square of the area as a quadratic in a² (you may want to substitute c for a²).
(b) By completing the square, determine the value of a for which the area is a maximum.
(c) Determine the value of b and the maximum area.
(d) What conclusion can you draw about the rectangle of maximum area?
(This is the first way, which I call the Algebra way)

(4) Divide the rectangle into 2 congruent triangles, using a diagonal. Draw a half
diagonal that intersects this diagonal.
(a) Write an inequality for the area of one of these triangles in terms of r alone. The inequality should be of the form Area ≤ _______.
(b) If you can achieve equality, then you have maximized the area of the rectangle! Find out when this occurs, and if it does, find the lengths of a and b. (The Geometry way).

(5) Method 3 - the Calculus way of course.
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Additional comments from DM:
(i) thanks, tc!
(ii) tc's geometric approach in (4) also suggests a connection to the famous AM-GM Inequality. Visit this link and see if you can make the connection. This is not obvious.
Hint: Apply the AM-GM to a² and b².