Pitot

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

October 11, 2026 Read

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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Definitions and elements

All points at the same distance from a center: the definition of the circle fits on a line, and yet it hides the number π. Who discovered The circle is one of the oldest forms in human history, from the wheels of Mesopotamia (c. 3500 B.C.) to stone monuments. The rigorous definition is from Euclid (definitions I.15 to I.18 of The Elements): a figure limited by a line whose points are all at the same distance from an interior point, the center. His third postulate guarantees that it is possible to draw a circle with any center and any radius. The first person to accurately calculate the ratio between contour and diameter was Archimedes (3rd century BC).

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