“It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.”

— Archimedes , book On the Equilibrium of Planes

Book 1, Proposition 14.
On the Equilibrium of Planes

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Greek mathematician, physicist, engineer, inventor, and ast… -287–-212 BC

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“In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side.”

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“The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides.”

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“If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity.”

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“In geometry the following theorems are attributed to him [Thales]—and their character shows how the Greeks had to begin at the very beginning of the theory—(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground”

Thomas Little Heath (1861–1940) British civil servant and academic

this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids
Achimedes (1920)

“The centre of gravity of a parallelogram is the point of intersection of its diagonals.”

Archimedes book On the Equilibrium of Planes

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“The centre of gravity of any cylinder is the point of bisection of the axis.”

Archimedes book The Method of Mechanical Theorems

Proposition presumed from previous work.
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“These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane… for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“Proposition 1. Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres.”

Aristarchus of Samos ancient Greek astronomer and mathematician

p, 125
On the Sizes and Distances of the Sun and the Moon (c. 250 BC)

“The classic instrument to measure drawn angles and to draw angles of a given measure is the protractor — essentially half a circular ring, subdivided by ray segments into 180 degrees. For reasons I was unable to find out, this instrument has recently been superseded by an isosceles right triangle — called geo-triangle, solid, transparant, made of plastic — with an angular division radiating from the midpoint of the hypotenuse to the other sides. Well, inside the triangle half a circle with the midpoint of the hypotenuse as its centre is indicated, and from the position of the degree numbers it becomes clear that it is the semicircle that really matters. One is inclined to say "an outrageously misleading instrument"…”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. 363

“To find the directions and quarters of the winds your method of procedure should be as follows. In the middle of the city place a marble amussium, laying it true by the level, or else let the spot be made so true by means of rule and level that no amussium is necessary. In the very center of that spot set up a bronze gnomon or "shadow tracker." At about the fifth hour in the morning, take the end of the shadow cast by this gnomon, and mark it with a point. Then, opening your compasses to this point which marks the length of the gnomon's shadow, describe a circle from the center. In the afternoon watch the shadow of your gnomon as it lengthens, and when it once more touches the circumference of this circle and the shadow in the afternoon is equal in length to that of the morning, mark it with a point. From these two points describe with your compasses intersecting arcs and through their intersection and the centre let a line be drawn to the circumference of the circle to give us the quarters of south and north…. and thus we have designed a figure equally apportioned among the eight winds.”

Vitruvius book De architectura

Source: De architectura (The Ten Books On Architecture) (~ 15BC), Book I, Chapter VI, Sec. 5-7

“It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.”

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