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Reading: Papus: the epilogue of Greek geometry

Further reading Leitura Papus: the epilogue of Greek geometry From the 3rd century BC, Rome established itself as a military power. It annexed Greece in 146 B.C. and Egypt in 31 B.C. Before that, in 47 B.C., a fire started in the Egyptian fleet, by order of Caesar, reached the library of Alexandria and consumed around 500 thousand texts. Alexandria would still be the cultural capital of the world for a long time, but Greek mathematics began to decline: the model of deductive geometry, without algebraic symbolism, showed signs of exhaustion, and the Romans only valued practical knowledge. In 529, the Greek schools of philosophy were closed, including Plato’s Academy. Plato, later artistic representation.Roman copy of a portrait of Silanion · Wikimedia Commons · Public domain The last great encouragement came from Papus of Alexandria (c. 300 AD). Its Mathematical Collection, in eight books, contains precious historical information, new demonstrations and original contributions. One of them is what is now called Guldin’s theorem: the volume of the solid generated by the rotation of a flat region around a straight line is equal to the area of ​​the region times the length of the trajectory of its center of gravity. Another is Papus’ theorem of projective geometry: if $A$, $B$, $C$ are on a straight line and $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ on another, the meetings of $AB^{\prime}$ with $A^{\prime}B$, $AC^{\prime}$ with $A^{\prime}C$ and $BC^{\prime}$ with $B^{\prime}C$ are collinear. Papus also owes the concept of focus and guideline of a conic. He strove to revive Greek geometry, but the forces of history were against him.

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Chapter exercises 8 — Notable points of the triangle

Exercises and resolutions Exercise 274 — Classify as true (V) or false (F) incenter is the center of the circle inscribed in the triangle. VF Verdadeiro. It is equidistant on three sides. circumcenter is the center of the circle circumscribed by the triangle. VF Verdadeiro. It is equidistant from the three vertices. incenter is internal to the triangle. VF Verdadeiro. The internal angle bisectors are inside the triangle. centroid is internal to the triangle. VF Verdadeiro. The medians are inside the triangle. orthocenter is internal to the triangle. VF Falso. In the obtuse angle it is external. circumcenter is internal to the triangle. VF Falso. In the obtuse angle it is external; in the rectangle, it is on the hypotenuse. centroid is the center of the circle inscribed in the triangle. VF Falso. This is incenter. ↻ Recomeçar

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orthocenter — heights

The three heights of a triangle also meet at a point, orthocenter. Interestingly, Euclid never proved this. Who discovered The The Elements do not demonstrate that the heights are concurrent. The first known published demonstration is by the Englishman William Chapple, in 1749. The most cited, however, is by Carl Friedrich Gauss (1810): he drew, for each vertex, the parallel to the opposite side, forming a larger triangle; the heights of the original triangle become the perpendicular bisectors of the larger one and therefore meet. The name orthocenter (from the Greek orthos, “straight”) was only created in the 19th century.

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circumcenter — perpendicular bisectors

The three perpendicular bisectors of a triangle meet at circumcenter, the center of the circle that passes through the three vertices. Who discovered The construction is the proposition IV.5 of The Elements of Euclid (c. 300 BC): “circumscribe a circle to a given triangle”. The meeting point of perpendicular bisectors is at the same distance from the three vertices, and therefore serves as the center. Euclid also observes where the center is: inside the triangle if it is acute, on the largest side if it is rectangular and outside if it is obtuse.

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incenter — angle bisectors internal

The three internal angle bisectors of a triangle meet at incenter, the center of the largest circle that fits inside it. Who discovered The construction is in proposition IV.4 of The Elements of Euclid (c. 300 BC): “inscribing a circle in a given triangle”. Euclid draws the angle bisectors from two angles, takes the meeting point and shows that it is the same distance from the three sides. As the third angle bisector also passes through this point, the three are competitors. A famous formula links the radius of this circle to the area: it is in the Metric of Heron of Alexandria (1st century AD), along with his formula for the area of ​​the three sides.

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centroid — medians

The three medians of a triangle meet at a point: centroid, where a triangular plate is balanced on the tip of a finger. Who discovered Archimedes of Syracuse (c. 287–212 BC) is the name linked to centroid. In On the balance of planes, he proved that the center of gravity of a triangle is on the straight line that connects a vertex to the midpoint of the opposite side — the median — and, therefore, at the meeting of the three medians. The word centroid is much more recent: it comes from the Greek barys, “heavy”, and was used in the 19th century by the German August Möbius, in the book The barycentric calculus (1827).

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