Plane Geometry

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).

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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.

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Exercises from the chapter 11 — Angles on a circle

Exercises and resolutions 377 Determine the value of the angle $x$ in the following cases: Ver respostaa) $35^{\circ}$ · b) $100^{\circ}$ · c) $60^{\circ}$ · d) $25^{\circ}$ · e) $50^{\circ}$ · f) $20^{\circ}$ 378 Determine the value of the arc $x$ in the following cases: Ver respostaa) $80^{\circ}$ · b) $30^{\circ}$ · c) $60^{\circ}$ 379 In the figures, calculate the value of $x$. Ver respostaa) $35^{\circ}$ · b) $10^{\circ}$

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Eccentric angles

When the vertex of the angle is inside or outside the circle, but not at the center or on it, the angle is eccentric — and its measure is an average of two arcs. Who discovered The formulas for eccentric angles are not stated in The Elements, but come directly from the proposition III.20 of Euclid (inscribed angle) combined with I.32 (external angle of a triangle). One of the most famous problems linked to them has an author: in 1471, the German astronomer Regiomontanus asked from what distance a statue on top of a pedestal is seen from the largest angle possible.

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Segment angle (semi-inscribed)

The angle between a chord and the tangent also measures half the arc. Hence the capable bow, used by navigators to escape rocks. Who discovered The measure of the segment angle is the proposition III.32 of the The Elements of Euclid, and the construction of the capable arc — the location of the points from which a segment is seen at a given angle — is the III.33. The best-known practical use has two names: the Dutch Willebrord Snellius (1617) and the French Laurent Pothenot (1692) solved the problem of discovering one’s own position from the angles under which three known points are seen, crossing two capable arcs.

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Inscribed angle

The inscribed angle measures half the arc that he sees. A special case — the angle inscribed in the semicircle is right — is attributed to Thales. Who discovered Greek tradition attributes to Tales of Miletus (6th century BC) the discovery that every angle inscribed in a semicircle is right. Diógenes Laércio says, citing the writer Pânfila, that Thales sacrificed an ox to celebrate the result. The general relationship — the central angle is twice that inscribed on the same arc — is the proposition III.20 of the The Elements of Euclid, and the case of the semicircle is the III.31.

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Central angle

The central angle has the apex at the center of the circle and measures the same as the arc it sees. With this idea, the size of the Earth was measured. Who discovered The relationship between central angle and arc is the basis of the proposition III.26 of The Elements of Euclid: in equal circles, equal central angles correspond to equal arcs. The most famous use of it is by Eratosthenes of Cyrene (c. 276–194 BC), director of the Library of Alexandria, who calculated the circumference of the Earth from a central angle.

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Congruence, addition and inequality of arcs

Measuring an arc in degrees is so natural today that we forget where it came from: from observing the sky. Who discovered The Babylonian astronomers divided the Sun’s path across the sky into 12 signs of 30 parts, totaling 360. The Greek Hyparchus of Nicaea (2nd century BC) took this measurement to geometry and created the first known chord table, which related each arc to the corresponding chord — the beginning of trigonometry. Claudius Ptolemy expanded the table in Almagest (c. 150 AD), with chords of half degrees. The comparison and sum of arcs of the same circle follow book III of The Elements by Euclid.

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Chapter exercises 10 — Circumference and circle

Exercises and resolutions 336 Determine the radius of the center circle $O$, given $AB = 3x - 3$ and $OA = x + 3$. Ver resposta$r = 12$ 337 The circle has a radius of 16 cm and the point $P$ is 7 cm from the center. Determine the distance between $P$ and the circumference. Ver resposta9 cm

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Tangent segments — circumscribable quadrilaterals

From a point outside a circle two tangents come out, and the two segments to the points of contact have the same length. Who discovered The equality of tangent segments is a direct consequence of proposition III.17 of The Elements of Euclid, which teaches how to draw the tangent through an exterior point. This gives rise to Pitot’s theorem: in a quadrilateral that has an inscribed circle, the sums of the opposite sides are equal. The French engineer Henri Pitot published it in 1725 (see the curiosity about him below). The reciprocal — if the sums of the opposite sides are equal, the quadrilateral has an inscribed circumference — was proven in the 19th century and is usually attributed to Jakob Steiner.

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Relative positions of two circles

Two circles can be secant, tangent, one inside the other or far apart. The distance between the centers decides which is the case. Who discovered The basic cases are in The Elements of Euclid (c. 300 BC): two distinct circles intersect at a maximum of two points (proposition III.10), and, when they touch, the point of contact is on the straight line that joins the centers (III.11 and III.12). A century later, Apollonius of Perga proposed the most famous problem on the subject: given three circles, draw a fourth tangent to all of them.

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Relative positions of straight line and circle

A straight line can cut a circle at two points, touch it at just one or pass far. The line that only touches is the tangent. Who discovered The tangent properties are in book III of The Elements by Euclid (c. 300 BC). The proposition III.16 shows that the straight line perpendicular to the radius, at its extremity, touches the circle at a single point, and the III.18 proves the opposite: the tangent is perpendicular to the radius at the point of contact. The word tangent comes from the Latin tangere, “to touch”.

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