Plane Geometry

Existence and uniqueness of parallel

A parallel passes through a point outside a straight line. That only one passes is the most discussed statement in the history of geometry. Who discovered The existence of parallelism is demonstrated by Euclid (proposition I.31). uniqueness depends on its fifth postulate, the one that many have tried to transform into a theorem. The form used today at school — “a single parallel passes through a point outside a straight line” — became known as Playfair’s axiom, after the Scotsman John Playfair, who popularized it in 1795, although Proclus had already stated it. In 1829, the Russian Nikolai Lobachevsky published a geometry in which infinite parallels pass.

October 11, 2026 Read

Parallel lines

Parallel lines never meet — and yet, in a perspective drawing, they appear to meet at the horizon. Who discovered The classic definition is from Euclid (c. 300 BC), in definition I.23 of The Elements: parallel lines are those that are in the same plane and, extended indefinitely in both directions, do not meet. The relationships between the angles formed by a transversal — alternate, corresponding, collateral — are the propositions I.27 to I.29. The philosopher Posidonius (1st century BC) proposed another definition, as straight lines always at the same distance; Proclus recorded the debate.

October 11, 2026 Read

Reading: Euclid and deductive geometry

Further reading Leitura Euclid and deductive geometry In 338 BC, Greece comes under Macedonian rule. Alexander the Great took Greek culture to the East and founded dozens of cities. One of them, Alexandria, in Egypt, would have an extraordinary role in the history of Mathematics. Ptolemy, who ruled Egypt after Alexander, created the Museum there, a research and teaching center maintained by the State. His library had close to 700 a thousand rolls of papyrus. Claudius Ptolemy, later artistic representation.Justus van Gent or Pedro Berruguete · Wikimedia Commons · Public domain One of the Museum’s first and most notable mathematicians was Euclid (c. 300 BC), about whose life almost nothing is known. It is said that when Ptolemy asked if there was a shorter way to learn geometry, he replied: “There is no real way in geometry.” Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain His fame rests on The Elements, in thirteen books: the oldest complete text of Greek mathematics to reach us. Since the first printed edition, in 1482, there have been more than a thousand editions. From five common notions, five postulates and some definitions, Euclid deduced 467 theorems. For two millennia, The Elements were the model of good mathematics. Over time, logical flaws were found (for example, the use of continuity hypotheses that were never stated), but they little diminish the greatness and influence of the work.

October 11, 2026 Read

Exercises from chapter 4 — Triangles

Exercises and resolutions Exercise 80 — Classify as true (V) or false (F) All isosceles triangle is equilateral. VF Falso. An isosceles can have a different base than the other sides. All equilateral triangle is isosceles. VF Verdadeiro. It has (at least) two congruent sides. A scalene triangle can be isosceles. VF Falso. Scalene does not have two congruent sides. Every isosceles triangle is an acute triangle. VF Falso. There are isosceles rectangles and obtuse angles. Every right triangle is scalene triangle. VF Falso. The isosceles rectangle (45°, 45°, 90°) is a counterexample. There are right-angled and isosceles triangles. VF Verdadeiro. Congruent legs. There is isosceles triangle obtuse angle. VF Verdadeiro. Obtuse vertex angle, such as 120°, 30°, 30°. Every acute triangle is either isosceles or equilateral. VF Falso. There are scalene acutangles, such as 50°, 60°, 70°. ↻ Recomeçar

October 11, 2026 Read

Inequalities in triangles

In every triangle, each side is smaller than the sum of the other two. The ancients already joked that “even a donkey knows this”. Who discovered The triangular inequality is the proposition I.20 of the The Elements of Euclid (c. 300 BC); the relationship between the largest side and the largest angle is in propositions I.18 and I.19. The idea that the segment is the shortest path between two points was intuitive long before; Euclides’ merit was to demonstrate it based on previous results.

October 11, 2026 Read

Congruence of triangles

When are two triangles “copies” of each other? There is no need to check the six elements: three well chosen ones are enough. Who discovered The oldest author-named criterion is ALA (one side and two adjacent angles). Proclus, citing Eudemus of Rhodes, says that Thales of Miletus (6th century BC) used it to calculate the distance of ships at sea. The demonstration of the cases LAL (proposition I.4), LLL (I.8) and ALA (I.26) is in The Elements of Euclid (c. 300 BC). In 1899, David Hilbert showed that the LAL case needs to be accepted as an axiom.

October 11, 2026 Read

Concept, elements and classification

The triangle is the simplest and most useful polygon: any other polygon can be divided into triangles. Who discovered Triangles appear in ornaments, constructions and area calculations since Egypt and Mesopotamia: the Rhind papyrus (c. 1650 BC) already presents problems about the area of triangular lands. The classification used to this day is from Euclid (c. 300 BC). In the definitions I.20 and I.21 of The Elements, he separates the triangles by the sides — equilateral, isosceles and scalene — and by the angles — rectangle, obtuse angle and acutangle.

October 11, 2026 Read

Exercises from the chapter 3 — Angles

Exercises and resolutions The calculator below does, step by step, the calculations with degrees, minutes and seconds for exercises 29 to 33. Try solving it first and use it to check. Enter the angles and choose the operation. It shows each step: 60″ becomes 1′ and 60′ becomes 1°. Ative o JavaScript para ver esta figura interativa. 29 Simplify the following measurements: a) $30^{\circ}70^{\prime}$ · b) $45^{\circ}150^{\prime}$ · c) $65^{\circ}39^{\prime}123^{\prime\prime}$ · d) $110^{\circ}58^{\prime}300^{\prime\prime}$ · e) $30^{\circ}56^{\prime}240^{\prime\prime}$ Ver respostaa) $31^{\circ}10^{\prime}$ · b) $47^{\circ}30^{\prime}$ · c) $65^{\circ}41^{\prime}3^{\prime\prime}$ · d) $111^{\circ}3^{\prime}$ · e) $31^{\circ}$

October 11, 2026 Read

Right, acute and obtuse angles — measurement

Why does a complete turn have 360 degrees, and not 100? The answer lies in Mesopotamian astronomy. Who discovered The names straight, acute and obtuse come from Euclid, who defines them in The Elements (definitions I.10 to I.12): straight is the angle equal to its supplementary adjacent; acute, smaller than the straight; obtuse, the biggest. The measurement in degrees has another origin, Babylonian astronomy, as explained in the curiosity “Why 360 degrees?”, below. The words minute and second come from the Latin translations of Claudius Ptolemy (2nd century AD).

October 11, 2026 Read

Congruence and comparison

Copying an angle and dividing it in half are classic constructions. Dividing it into three equal parts, however, has challenged mathematicians for more than two thousand years. Who discovered The two constructions are in The Elements of Euclid (c. 300 BC): the proposition I.9 divides an angle in half (a angle bisector) and the I.23 copies a given angle onto another straight line. The trisection of the angle with a ruler and compass — dividing any angle into three equal parts — was proven impossible in 1837 by the Frenchman Pierre Wantzel, then just over 20 years old.

October 11, 2026 Read

Angle: definitions

Two rays with the same origin form an angle. And the first theorem about angles even has a named discoverer. Who discovered According to Proclus (5th century AD), who cited the History of Geometry by Eudemus of Rhodes, it was Thales of Miletus (c. 624–546 BC) who first realized that angles opposite the vertex are equal; Euclid only later demonstrated it “scientifically”, in proposition I.15. The definition of angle as “inclination of two lines” is in The Elements of Euclid (definition I.8).

October 11, 2026 Read

Introduction: convexity and half-planes

A straight line divides the plane into two sides. A figure is convex when it has no “indentations”. The two ideas go together. Who discovered The first to carefully define a convex figure was Archimedes of Syracuse (c. 287–212 BC), in the work On the sphere and the cylinder, where he uses curves “concave to the same side” to compare perimeters. The separation of the plane by a straight line was formalized by Moritz Pasch in 1882: his axiom says that a straight line that enters a triangle on one side must exit on the other. At the end of the 19th century, Hermann Minkowski made convexity an area of mathematics.

October 11, 2026 Read