Why is construction required in geometry?

Question

I don't know if this is a good question to ask but I need some advice, anything is appreciated.

(Contruction here, doesn't imply geometric construction tools one)

Q1:In solving problems in geometry, we often have to make a construction. Making a construction, does not create anything new. Then why can not we solve problems without any construction?

Why do we need to make constructions, if they don't add any new condition to the problem?

Q2: How do you motivate towards a construction? I have a hard time at hitting at the right construction, so is there any way to know what to do, in order to arrive at the proof or is it again just experience?

(It would help if you answer in an olympiad perspective.)

Answer 1

2500 years ago, when the Greeks were first formalizing their systems of mathematical thinking, visual "geometric" figures were the means to communicate everything. There was no algebraic or logical notation. When we learn geometry, we tend to start where they started, and as they started, with constructs.

In 2D or 3D Euclidean geometry, because of the correspondence of the subject to our sensory world, it is easy and natural to use drawings to convey a proof, to organize our thinking, or to recognize patterns or possibilities in the situation. But a construct is just a visual form of a proof. The proof exists independently of the visual construct, and could be developed entirely without it.

Answer 2

Geometric construction appears to be a game of sorts, but it serves a deeper purpose: when done using unmarked straightedge and compasses as instructed by Euclid, it amounts to defining the object via Euclid's axioms, through the point-line relationships and congruences ensured by the construction.

Such definitions then form the basis for proofs. For instance once the Greeks discovered (or perhaps learned from predecessors, we don't really know) constructions for the regular pentagon, they (and we, today) could use those constructions to define it and prove various properties entirely from Euclid's axioms. But they could not achieve such a definition for the regular heptagon, so proofs involving it necessarily had a gap (having to assume the polygon somehow exists) if they were to be done only with the Euclidean axioms.

Answer 3

The answers you got so far have misinterpreted your question.

Why do we need to make constructions, if they don't add any new condition to the problem?

In theory, no construction is needed for a very wide class of Euclidean geometry problems, namely whose solutions can be obtained using Tarski's axiomatization. The reason is that every such geometry problem can be expressed using polynomial equations, and the solution can be proven by purely logical deduction (including algebraic manipulation) from those axioms, and in fact there is a computer program that can systematically find the solution in every case.

In practice, we cannot easily solve geometry problems using this method, since the degree of the relevant polynomial roughly doubles with every geometric construction step that involves a circle, and worse still the time taken by such a (systematic) program can be exponential in the total degree of the relevant polynomial. This makes a purely mechanical solution infeasible in many cases.

Instead, in many geometry problems (especially in olympiads), one can find a solution that is comprised of a series of independent steps, each of which has a simple proof. But sometimes, such solutions pass through geometric constructions that are not needed in stating the original problem. If the shortest solution passes through some such intermediate construction, then you may feel that the problem can be solved only if you make such constructions. In theory (explained above), this is false. But in practice, proofs without such constructions can be so much longer or unsatisfying that it is fair to say that the intermediate constructions are important.

For example, to prove that every triangle △ABC with AB = AC satisfies ∠ABC = ∠ACB, the easiest proof is to let M be the midpoint of BC and prove that △ABM ≡ △ACM, and hence that ∠ABM = ∠ACM. In theory, you can prove the result purely algebraically without constructing any new points, but it may feel uninspiring.

For example, use complex numbers a,b,c for points A,B,C, and let x = a−b and y = a−c, with conjugates x',y'. If abs(x) = abs(y), then x·x' = y·y', so arg((c−b)/(a−b)) = arg((y−x)/x) = arg(y/x−1) and arg((b−c)/(a−c)) = arg((x−y)/y) = arg(x/y−1), and we can check that arg(y/x−1) + arg(x/y−1) = 0, because (y/x−1)·(x/y−1) = 2+x/y+y/x is real since x'/y' = y/x and y'/x' = x/y. Therefore ∠CBA + ∠BCA = 0 (in directed angles).

Answer 4

TL;DR under the separating line

This is not quite what you ask for, but a similar question can be made in analysis as well, i.e. why do we sometimes add $0$ to an expression (usually in the form of $0=a-a$) when, technically and literally speaking, it adds nothing.

The answer to that question is that it provides an intermediary step by reducing the problem to parts that are easier to solve, or already have been solved. For example, one may want to prove the convergence of a sequence of functions $f_n \to f$ in some norm, i.e. we which to prove that $\lVert f_n-f \rVert \to 0$. Sometimes that is not easy. However, suppose that each $f_n$ is close to some $g_n$ (e.g. $\lVert f_n-g_n \rVert < \varepsilon$ or $\lim_n \lVert f_n-g_n \rVert = 0$) that converges to some $g$ that happens to be close to $f$, i.e. $\lVert g-f \rVert < \varepsilon$, then we can "add zeroes" to get \begin{align} \lVert f_n-f \rVert &= \lVert f_n+0+0-f \rVert \\ &= \lVert f_n +(g_n-g_n)+(g-g)-f \rVert \\ &= \lVert (f_n -g_n) + (g_n-g)+(g-f) \rVert \\ &\le \lVert f_n-g_n \rVert + \lVert g_n-g \rVert + \lVert g-f \rVert, \end{align} so we can conclude that $\limsup_n \lVert f_n-f \rVert \le 2\varepsilon$ (or $\le \varepsilon$), and if such $g,g_n$ can be found for any $\varepsilon>0$, then we can ultimately conclude that $f_n\to f$ as wanted. In fact, this type of technique is widely used in analysis.

You can see that even though we "add zeroes" to the term we are interested in, we did not "do nothing". What we did was recognizing that $0$ can be written in a way that benefits us and simplifies the problem. So, by analogy, to answer your questions:

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A1: Constructions, even though they do not provide any more information, they help us visualize and (hopefully) reduce our problem to a simpler one (or break it down to many simpler ones). It is not needed, per se, so if you can find a way to solve a problem without using any construction, then I don't see a problem with that either.

The argument "Why do we do this when it technically doesn't add anything new?" can be carried all the way to "Why do we write down theorems when they add nothing new to the axioms system?". We humans are just not that good at going straight to all the conclusions that can be derived from given information, so intermediary steps (doing constructions, adding $0$ etc.) are helpful, or even necessary.

A2: It comes with experience and practice. I myself am horrible at solving Olympiad problems in geometry, but my experience with analysis does help me figuring out when to "add zeroes" to my equations/inequalities. There is a saying I don't quite remember where I first heard it from, but it goes like this:

"It takes knowledge to know that $\cos(\theta)^2+\sin(\theta)^2$ can be simplified to $1$, but it takes wisdom to know when to write $1$ as $\cos(\theta)^2+\sin(\theta)^2$."

I think this applies very well here too.

Answer 5

Can we prove e.g., the Pythagoras theorem without a construction?

A construction is necessary to assemble relevant previous results implicitly discarding irrelevant results step by step, to synthesise proof towards a new result or conclusion.

Even so, often there is more than one construction method employed.