Nearly everything in Euclidean geometry comes down to a divide-and-conquer approach:

- Reduce the question to a question about triangles.
- Use our extensive knowledge of triangles to answer it.

Several people have mentioned coordinate geometry, but ultimately that, too, comes down to our basic ideas about triangles. Without establishing those we haven't distinguished Euclidean geometry from, say, hyperbolic geometry, so we wouldn't be able to conclude anything nontrivial about our coordinates.

If there's a problem here it's not the material, but perhaps that students see this as a grab-bag of disconnected facts instead of the basis of all the math they're going to use in the future. Every single math class from here on out relies on a strong understanding of triangles, and that becomes more and more pronounced as things progress:

In **calc 1** they'll learn to show that
$$\lim_{x \to 0} \dfrac{\sin(x)}{x} = 1$$
using a geometric argument and the squeeze theorem, in order to determine that
$$\dfrac{d}{dx} \sin(x) = \cos(x).$$

In **calc 2** they'll need a solid grasp of trig so that they can take integrals like
$$\int \dfrac{1}{1 + x^2} \, dx = \tan^{-1}(x).$$

In **vector calculus** they'll need to be able to convert between cartesian, polar, and spherical coordinates and understand why, say,
$$dx \, dy \, dz = r^2 \sin \phi \, dr \, d\phi \, d\theta$$
and moreover they'll need to be able to parametrize a given region in three-dimensional space in, say, spherical coordinates.

In **differential equations** (or possibly a physics or engineering class) they'll need to grasp the Fourier transform.

In **linear algebra** they'll learn about inner products in finite dimensions.

1To give all students a chance to succeed in university, should they choose that path. Many large American universities assume students are at least familiar with many of those topics, even in most beginning courses. – Chris C – 2015-01-04T16:31:08.217

4@ChrisC That's just pushing the question one level farther back. Also, many, many large universities are not in the USA. – David Richerby – 2015-01-05T08:27:58.793

I saw this on the "Hot Questions" list and couldn't help thinking of this: https://www.youtube.com/watch?v=TdRqQReggOI

– Josh – 2015-01-06T03:55:52.9871I am not going to say why they are so prevalent. But I think, if you can relate triangles to their uses, it would be very helpful and interesting for the students and for you too. – Karthick S – 2015-01-06T10:57:52.973

4Because they represent the middle class in

Flatland. – vpipkt – 2015-01-06T15:38:21.447But seriously as an engineering student at the bachelor's level, this was truly essential background. After a few years in the work force, I echo @Keith 's comment about spherical coordinates. – vpipkt – 2015-01-06T15:39:56.140

It may be interesting to take Euclid's

Elements, the books on plane geometry, and see what portion of those propositions are on triangles. Do you think it would be substantially less than the one-third in the question? – Gerald Edgar – 2015-05-03T13:15:30.730It is to be sure to induce a greater sense of wonder when learning about non Euclidean geometry.. – yngabl – 2016-09-26T11:48:51.540