Plane Geometry

Rectangle, rhombus and square

Diagonals tell a lot about a quadrilateral. In the rectangle, they are equal; in the rhombus, perpendicular; in the square, both. Who discovered The most famous document on the diagonal of the square is the Babylonian tablet YBC 7289, from around 1800–1600 BC, today at Yale University. It shows a square with diagonals and a value of $\sqrt{2}$ with an error of less than one millionth. The properties of the diagonals of the rectangle and the rhombus follow from the theorems on parallelograms of The Elements of Euclid.

October 4, 2026 Read

Properties of parallelograms

Opposite sides equal, opposite angles equal, diagonals that cut in half. The parallelogram is the figure of forces and articulated mechanisms. Who discovered The first properties are in the proposition I.34 of the The Elements of Euclid (c. 300 BC): in a parallelogram, the opposite sides and angles are equal, and the diagonal divides it into two parts of the same area. Outside of geometry, the parallelogram gained fame with the parallelogram rule of forces, used by the Flemish Simon Stevin (1586) and enunciated by Isaac Newton in the Principia (1687): two forces acting at a point are equivalent to the diagonal of the parallelogram formed by them.

October 4, 2026 Read

Properties of trapezoids

The trapezoid has a pair of parallel sides — and is the shape hidden in many ramps, rooftops, and pockets. Who discovered The modern meaning of the word comes from Posidonius (1st century BC) and Proclus (5th century AD): he records that Posidonius separated quadrilaterals with a pair of parallel sides from those that had none. For Euclid, trapezoid was any quadrilateral that was not a square, rectangle, rhombus or parallelogram. The area of the trapezoid — the average of the bases times the height — appears in the works of Heron of Alexandria (1st century AD).

October 4, 2026 Read

Notable quadrilaterals — definitions

Trapezoid, parallelogram, rectangle, rhombus and square. The names are old, but what each means has changed over the centuries. Who discovered Squares and rectangles already appear in Babylonian tablets and in Egyptian area problems. The definitions came from Euclid (definition I.22 of The Elements), who distinguished square, oblong, rhomboid and rhomboid and called trapezoids all other quadrilaterals. The word parallelogram appears for the first time in Euclid’s proposition I.34. The inclusive classification used today, in which every square is a rectangle, was adopted at the end of the 19th century, as explained in the curiosity “Definitions changed over time”, below.

October 4, 2026 Read

Quadrilateral — definition and elements

Four sides, four vertices, two diagonals. Quadrilaterals are in lots, screens, tables and windows. Who discovered Calculating the area of land on four sides was the task of the scribes of Egypt and Mesopotamia. Inscriptions from the temple of Edfu, in Egypt (2nd century BC), record a rule for the area of a four-sided plot: multiply the average of two opposing sides by the average of the other two. The rule is only exact for rectangles and overestimates the area in other cases. The geometric definition — figure limited by four straight lines — is from Euclid (definition I.19 of The Elements).

October 3, 2026 Read

Exercises from chapter 6 — Perpendicularity

Exercises and resolutions 189 If $\overline{AH}$ is height relative to the $\overline{BC}$ side of $\triangle ABC$, determine $\hat{B}$ and $\hat{C}$ in the cases: Ver respostaa) $\hat{B} = 70^{\circ}$ e $\hat{C} = 40^{\circ}$ · b) $\hat{B} = 60^{\circ}$ e $\hat{C} = 40^{\circ}$ 190 In each case below we have a base isosceles triangle $\overline{BC}$. Determine the angle of the base. Ver respostaa) $65^{\circ}$ · b) $61^{\circ}$

October 3, 2026 Read

Projections and distance

The shadow of an object with the Sun overhead is an orthogonal projection. Engineers turned this idea into a drawing method still used today. Who discovered The idea of projecting figures onto a plane is old — sundials already read the projected shadow of the gnomon. The person who transformed it into a method was the Frenchman Gaspard Monge (1746–1818), creator of descriptive geometry. Monge represented objects by their orthogonal projections on two perpendicular planes, the plan and the elevation, and thus solved construction problems on paper. The distance from a point to a straight line as the length of the perpendicular is a classic result of The Elements of Euclid.

October 3, 2026 Read

Existence and uniqueness of the perpendicular

From a point outside a straight line, how can you lower a perpendicular to it? This is one of the few old issues with a registered author name. Who discovered Proclus attributes to Enopides of Chios (5th century BC) the first study of the problem of drawing a perpendicular to a straight line through a point outside it; Oenopides called this line “gnomonic”, in reference to the gnomon, the vertical rod of sundials. The classic solutions are in The Elements of Euclid: proposition I.11 raises the perpendicular by a point on the line, and I.12 lowers it by a point outside it.

October 3, 2026 Read

Definitions — right angle

Walls, squares, sheets of paper: the built world is full of right angles. Who discovered The right angle was used in practice by Egyptians, Babylonians, Indians and Chinese long before it was defined. The Babylonian tablet Plimpton 322 (c. 1800 B.C.) lists sets of numbers that form right triangles, such as 45, 60, and 75 — the triangle 3-4-5 magnified 15 times —, and the Chinese text Zhoubi Suanjing brings a famous figure about the triangle 3-4-5. The rigorous definition is from Euclid (definition I.10): when a straight line meets another forming equal adjacent angles, each of them is straight, and the straight line is perpendicular to the other.

October 3, 2026 Read

Reading: The fifth postulate

Further reading Leitura The fifth postulate The parallel postulate is the fifth postulate of The Elements. Since Antiquity, many mathematicians have distrusted it: it seemed more like a theorem than an evident truth, and several generations tried to demonstrate it from the other four. All attempts failed — and in the 19th century, Gauss, Bolyai and Lobachevsky realized why: by changing the fifth postulate to “through a point several parallels pass”, one obtains a geometry as consistent as Euclid’s, hyperbolic geometry. In it, the sum of the angles of a triangle is less than 180°. Euclid’s postulate is not a logical necessity; It is a choice that defines the geometry of our daily lives. Carl Friedrich Gauss.Christian Albrecht Jensen (1792–1870) · Wikimedia Commons · Public domain János Bolyai, imaginary portrait by Ferenc Márkos (2012).Ferenc Márkos · Wikimedia Commons · CC BY-SA 3.0 Nikolai Lobachevsky, portrait by Lev Kryukov.Lev Kryukov · Wikimedia Commons · Public domain Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain

October 3, 2026 Read

Exercises from chapter 5 — Parallelism

Exercises and resolutions 130 Since the line $a$ is parallel to the line $b$, determine $x$ in the cases: Ative o JavaScript para ver esta figura interativa. Ver respostaa) $50^{\circ}$ · b) $60^{\circ}$

October 3, 2026 Read

Consequences in the triangle

The sum of the interior angles of any triangle is 180°. It is perhaps the best-known geometry theorem after Pythagoras'. Who discovered Proclus, citing Eudemus of Rhodes, attributes to the Pythagoreans (6th–5th centuries BC) the first general demonstration that the angles of a triangle add up to two right angles. The commentator Gemino adds that the ancients had proven the fact separately for equilateral triangles, isosceles and scalenes. The version that has come down to us is the proposition I.32 of the The Elements of Euclid, which also proves that the external angle is equal to the sum of the two non-adjacent internal angles.

October 3, 2026 Read