Angle Bisector

Constructions with ruler and compass

A ruler without marks and a compass: with just these two tools, the Greeks constructed midpoints, perpendiculars and parallels with total precision. Who discovered The classical constructions are in The Elements of Euclid (c. 300 BC): the perpendicular bisector and the midpoint (proposition I.10), the angle bisector (I.9), the perpendicular by one point (I.11 and I.12) and the parallel by one point (I.31). The rule of using only a ruler without marks and a compass comes from Euclid’s first three postulates: draw a straight line through two points, extend it and draw a circle with any center and radius.

October 12, 2026 Read

Reading: Legendre: towards a rigorous and didactic geometry

Further reading Leitura Legendre: towards a rigorous and didactic geometry In ancient Greece, axiom seemed to mean a truth common to all fields of study, and postulate, a truth of a specific field. Euclid assumed five postulates — among them, that from any point a straight line is drawn to any other, that every limited straight line extends, that with any center and radius a circle is drawn, that right angles are equal, and the famous fifth, that of parallels — and five axioms, such as “things equal to the same thing are equal to each other” and “the whole is greater than the part”. From then on, The Elements rattle off theorems without exercises or applications: a text that is not very didactic by modern standards. Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain In 18th-century France several books appeared to make Euclid more accessible, but many sacrificed rigor. A notable exception was Elements of Geometry by Adrien-Marie Legendre (1752–1833). Legendre shined in his research — in elliptic functions and number theory, in which he completely enunciated the law of quadratic reciprocity — and his geometry book, which reconciled rigor and didactic, had twenty editions in France during the author’s lifetime. In 1809 it was translated in Rio de Janeiro and adopted at the Royal Military Academy, created the following year. For about forty years, Legendre tried to prove the fifth postulate from the others — an impossible task, as we now know — but even in these attempts he showed great ingenuity. Adrien-Marie Legendre, caricature by Julien-Léopold Boilly.Julien-Léopold Boilly · Wikimedia Commons · Public domain

October 11, 2026 Read

Exercises from chapter 12 — Thales' Theorem

Exercises and resolutions 413 Determine the value of $x$ in each case below, with $r$, $s$ and $t$ parallel lines. Ver respostaa) 3 · b) 12 · c) 15 · d) 6 414 In the figures, the lines $r$, $s$ and $t$ are parallel. Determine the values of $x$ and $y$. Ver respostaa) $x = \dfrac{10}{3}$ · b) $x = \dfrac{25}{6}$ · c) $x = \dfrac{10}{3}$ e $y = \dfrac{18}{5}$

October 11, 2026 Read

angle bisectors Theorems

The angle bisector of an angle of a triangle divides the opposite side into parts proportional to the other two sides. Who discovered The internal angle bisector theorem is the proposition VI.3 of the The Elements of Euclid (c. 300 BC). The one with external angle bisector is not in the Greek text: it was added as “proposition A” of book VI by the Scotsman Robert Simson, in his edition of The Elements of 1756, which served as the basis for teaching geometry in the United Kingdom for more than a century. The two angle bisectors are linked to the circle of Apollonius, studied by Apollonius of Perga (3rd century BC).

October 11, 2026 Read

Thales' theorem

A bundle of parallel lines cuts two transversals into proportional parts. It is with this idea that a board is divided into equal parts without doing any math. Who discovered The result is called Tales of Miletus (6th century BC), associated by tradition with the measurement of the height of the Great Pyramid by shadows (see the curiosity about it at the opening of the chapter). The demonstration we know is from Euclid, in proposition VI.2 of The Elements (c. 300 BC), and uses areas of triangles. The name “Tales’ theorem” for this result is a tradition in countries such as Brazil, Portugal, France, Spain and Italy; in German and English, the same name refers to the angle inscribed in the semicircle.

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