Content
Presented in this section is a collection of the many ways to construct a Euclidean two-dimensional plane triangle.
For this particular text I have settled on the spirit of intuition, clarity, search for a solution and deductions from first principles. The rigor of proofs which, however useful and needed, tends to kill the above objective and, as such, the proofs are left up to the reader.
Having said that I would like to make two more important comments about correctness and style.
Even though I will not be showing the proofs the constructions must nonetheless be mathematically correct. It means that for any given construction it should always be possible to come up with a proof of its correctness.
In his book "Elements" Euclid carries out a large number of straight edge and compass constructions. The nature of Euclid's work, however, is very hierarchical and sequential. Within any current construction Euclid is extremely careful to use only those geometric properties that have been proven to be true previously, up to this point.
We shall take it that we are not bound by such a requirement and we will assume that all of the geometric knowledge acquired so far is available to us at once. This knowledge includes all the properties proven to be true in Euclid's "Elements" plus all the relevant geometric properties and theorems that were discovered since.
As usual, we will be referring to Euclid's "Elements" in an abbreviated form:
$$B3P15$$for example, stands for Book \(3\) Proposition \(15\) and so on.
We will be using \(g\) to record the construction steps. Read more about this small and simple language in the \(g\) section.
We will not be going into the details of the Elementary Constructions or ECons. Instead, we will use them as atomic building blocks recorded with \(g\) shortcuts.
ECons show how to construct perpendiculars, parallel straight lines, perpendicular and angle bisectors, tangents, copies of an angle and a line segment, sum, difference, product, ratio and radical lengths of line segments and so on. Refer to ECons section for their full list.
Approaches
The two most valuable approaches for solving the straight edge and compass construction problems that we will be using are Reverse Order and Loci approaches.
With Reverse Order we always assume that the required construction has been carried out somehow. How - we do not know but we assume that it was done and it was done correctly. Next, using the accomplished construction as a starting point we will attempt to find useful observations that will allow us to carry out the construction in the normal order with the objects given.
With loci approach we will use the intersection of two or more geometric objects to locate useful points. By definition a locus is a geometric set of all the points and only those points that satisfy some number of conditions or constraints. An intersection of two or more of such loci produces a point that simultaneously belongs to multiple curves and thus simultaneously possesses multiple qualities which can be profitably interpreted as needed.
Notation
\(a\), \(b\), \(c\) will designate the lengths of the sides or simply "the sides" of a triangle.
\(\angle A\), \(\angle B\), \(\angle C\) will designate the angles of a triangle opposite to the corresponding sides: \(\angle A\) is opposite to \(a\), \(\angle B\) is opposite to \(b\), \(\angle C\) is opposite to \(c\).
By default the objective of all the problems is to construct an acute scalene triangle unless noted otherwise. Triangles with special properties - right, obtuse, isosceles, equilateral - that require drastically different construction methods if any will be treated separately:
| sss | sas | asa | aas | ssa | saSOs | sSOsa |
| saDOs | sDOsa |
\(\blacksquare\)