a geometric construction
Given:
Define a geometric construction as an object that can be created using only a compass and a straightedge. Mathematicians have shown that it is not possible to:
1. Geometrically construct a square with an area equal to that of a given circle.
2. Use a geometric construction to trisect an arbitrary angle.
The proofs of these two theorems require abstract algebra.
Task:
Summarize the history of these two problems.
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Solution Preview
I am really not sure the level of detail your teacher will want, but I will try to identify what I consider to be some key points in the history of these problems.
Squaring the circle (also called Quadrature of the Circle):
Solving this problem using only a straightedge and a compass hinges on knowing the value of pi. Some events in the history of the problem are:
1. Problem is mentioned in the Egyptian Rhind papyrus dating back to 1800 B.C.. Pi is approximated as 256/81 and the area of the circle is given by 64/81 d^2.
2. Many ...
Solution Summary
This solution strongly emphasizes a geometric construction.