Geometry 4: Right Triangles within Circles

This series is about how the megalithic, which had no written numbers or arithmetic, could process numbers, counted as “lengths of days”, using geometries and factorization.

This lesson is a necessary prequel to the next lesson.

It is an initially strange fact that all the possible right triangles will fit within a half circle when the hypotenuse equals the half-circles diameter. The right angle will then exactly touch the circumference. From this we can see visually that the trigonometrical relationships, normally defined relative to the ratios of a right triangle’s sides, conform to the properties of a circle.

A triangle with sides {3 4 5} demonstrates the general fact that, when a right triangle’s hypotenuse is the diameter of a circle, the right angle touches the circumference.

Planets and their orbits are all somewhat circular and, by the time of Aristotle, circular motion became associated with a higher domain called eternity, perhaps because the rotation and orbital cycles of planets never seem to run down with time. Life on earth is less obviously regulated by circular motions as all cycles and dynamic systems are founded on the principle of eternally returning to start again. Hence King Solomon said, “There is nothing new under the Sun” [Ecclesiastes 1:9].

In placing right triangles into the diameter of a circle equal to their hypotenuse (or longest side) we therefore return them to their perfect nest, from which more can be seen about them. Readers of articles about Le Manio and other megalithic landscapes around Carnac, will be aware that the astronomy there, between 5000 and 3000 BC, exploited the single, double, triple, and quadruple square rectangles. The first three gave the extremes of the Moon and Sun on the horizon (figure 1) and the second pair could proportionately count two different time cycles of the sun and moon such as their years. A 3 by 4 rectangle (also two right triangles, sides {3 4 5}) gave the solsticial extremes of the Sun on the horizon and this may be why the astronomers chose Carnac as the latitude where that occurred [Heath, 2014].

Figure 1 How the simplest rectangular forms defined astronomy at Le Manio’s Quadrilateral.

Around Carnac, many monuments suggest the use of such rectangles in their design; in which one diagonal (i.e. hypotenuse) or side is east-west, north-south, or at the extreme angles of the sun or moon on the horizon, equally in each of the four quadrants relative to the east-west axis.

A right triangle can be combined with a copy of itself, when rotated through 180 degrees, to form a rectangle. Thus any square or rectangle can, like any right triangle, use two opposite corners as the diameter of a circle – since all its corners are right angles.

We can take the simplest Pythagorean {3 4 5} triangle to see how right triangles relate to rectangular and circular structures (figure 2). However, all right triangles are defined by the circle equal in diameter to their hypotenuse. Each triangle can be in just four reflex forms within the circle.

Figure 2 The two in-rectangles containing 4 possible triangles on that diameter.

The hypotenuse therefore defines the out-square of the circle (figure 3)

Figure 3 The out-square of the circles of fig. 1, showing the areas of the two smaller sides, 9 and 16.

The out-square is naturally divided into two inner areas equal to the squares of the shortest and, around that, the next shortest side. Pythagorean triangles maintain their integer side lengths exactly because of this compatibility in their squared areas.

In figure 2 and 3 we see a circle and square of diameter  equal to the hypotenuse of right triangle {3 4 5}. Each triangle (fig. 2) has an area of 6 (3 x 2) and all four triangles have a combined area of 24, one less than the area of the square side 5 (fig 3). Tessellated, the four triangles leave one square in the center (fig. 4)

Figure 4 The four triangles tessellated within the 5-square, leaving one unit between them.

This central unit is the difference between the smaller sides (4 – 3 = 1). The product of the smaller sides (4 x 3 = 12), times 2, equals 24. Subtracting this from the square of the longer side leaves 25 – 24 = 1 square unit area.

Consider the next Pythagorean triangle, 5-12-13: Firstly, its area is 30, so times 4 the area equals 120. Secondly, the difference between the smaller sides is 7 which, squared, equals 49. And 120 + 49 = 169, which is 13 squared (figure 5).

Figure 5 The rapid growth of central area as triangles get more acute, here with 2nd Pythagorean triangle {5 12 13}.

If the smaller sides were equal, the geometry would be one quarter of a square that touches the center, and there would be no central region. If one side equaled the hypotenuse, there would be no third side and hence zero area. In between these boundary conditions there are differential side lengths and hence the square of the hypotenuse will be larger than the squares of the smaller sides and equal to them.

In the next lesson, we will look in the same way at the non-Pythagorean triangle found at Le Manio (see fig. 1 above – far right), which has sides {3 12} which makes it fit within a quadruple-square’s rectangle. This geometry is also found fitting within a {5 12 13} geometry, the 2nd Pythagorean triangle of figure 5; if an intermediate hypotenuse is struck to the 3rd side (= 5), at 3 units above right angle (figure 6).

Figure 6 The relationship of 2nd Pythagorean triangle and the 12 lunar months in the Lunar Year and the 12 .368 lunar months in the solar year, also then a quadruple-square rectangle. The larger triangle allows a length important to tracking eclipses on the smaller (Robin Heath, 1998). Le Manio preceded the station stones of Stonehenge by at least 1000 years.