Abstract
This chapter develops a set of tools which will allow us to start a systematic study of the metric aspects of Plane Euclidean Geometry; generally speaking, the central problem with which we shall be concerned here is that of comparing ratios of lengths of line segments. Among several interesting and important applications, the most prominent ones are the theorems of Thales and Pythagoras, which will reveal themselves to be almost indispensable hereafter. We also present a series of classical results, among which we highlight the study of the Apollonius circle and the solution of the Apollonius tangency problem, the collinearity and concurrence theorems of Ceva and Menelao, and some of the many theorems of Euler on the geometry of the triangle.
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Notes
- 1.
For the necessary background, see sections 7.1 and 7.2 of [5], for instance.
- 2.
After Thales of Miletus , Greek mathematician and philosopher of seventh century bc, the first of Classical Antiquity.
- 3.
- 4.
Pythagoras of Samos was one of the greatest mathematicians of Classical Antiquity. The theorem that bears his name was already known to babylonians, at least two thousand years before he was born; nevertheless, Pythagoras was the first one to prove it. It is also attributed to him the first proof of the irrationality of \(\sqrt {2}\).
- 5.
It can be shown that the set of points of tangency is actually equal to \(\mathbb Q\). For a proof, we refer the reader to [21].
- 6.
James Sylvester , English mathematician, and Tibor Gallai , Hungarian mathematician, both of the twentieth century.
- 7.
- 8.
Claudius Ptolemy , Greek astronomer and mathematician of the second century ad, gave great contributions to Euclidean Geometry. Ptolemy is mostly known for his work as an astronomer, especially for having proposed the (wrong) Geocentric Theory, according to which the Earth occupied the center of the Universe. Such a theory was accepted as a dogma by the Catholic Church for some 1400 years, and was the ultimate responsible for the judgement of Galileo Galilei by the Saint Inquisition.
- 9.
The Swiss mathematician Leonhard Euler , who lived in the eighteenth century, is generally accepted to be one of the mathematicians who most published relevant works. His contributions vary, impressively, from Geometry to Combinatorics (in which he created Graph Theory), passing through Number Theory and Physics. In each one of these areas of Mathematics there is at least one celebrated Euler’s theorem.
- 10.
Some sources attribute the discovery of the facts listed in items (a) and (b) to the French emperor Napoleon Bonaparte, whereas others suggest that this is apocryphal.
- 11.
Pierre Simon de Fermat , French mathematician of the seventeenth century.
- 12.
Such a concept will be taken up again, in a thorough way, in Sect. 9.2.
- 13.
Joseph Gergonne , French mathematician of the nineteenth century.
- 14.
Christian Heinrich von Nagel , German mathematician of the nineteenth century.
- 15.
Pappus of Alexandria , Greek mathematician of the fourth century. For another proof of Pappus’theorem, see Problem 8, page 329.
- 16.
Gaspard Monge , French mathematician of the eighteenth and nineteenth century, gave several important contributions to Geometry, particularly to Differential Geometry.
- 17.
Jean Victor Poncelet , French mathematician of the nineteenth century. For a proof of this general version of Poncelet, see Chapter 4 of the beautiful book [23].
- 18.
Such a name is due to the fact that, in the notations of the proof of Theorem 4.35, P ∈ e if and only if \(\sqrt {\overline {PO_1}^2+R_2^2}=\sqrt {\overline {PO_2}^2+R_1^2}\), so that we have equal radicals.
- 19.
The attentive reader has certainly noticed that, according to Problem 9, page 37, if H and O are the orthocenter and the circumcenter of a triangle ABC, respectively, then the pedal circle of {H, O} with respect to ABC is precisely the nine-point circle of ABC.
References
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Caminha Muniz Neto, A. (2018). Proportionality and Similarity. In: An Excursion through Elementary Mathematics, Volume II. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77974-4_4
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