The Triumphant Triangle
As mentioned last time, in order to "square up" a patio, you have to make a triangle. Of course, this shape is named by the number of angles it has. After a straight line between two points, the next thing is getting to a third point. The shortest path between the three points forms a three-sided enclosure with three angles: the "tri-angle."
The prefix "tri" bypasses the introduction of the "th" (theta) that replaced the original "t" (tau) in English. The vowel sound changes depending on the context and language, but the original Indo-European "trei" is preserved in the Greek treis and its cognates in Western languages. From all I could find out, its meaning has always been "three" (2 + 1).
Though there seems to be a dualism in much of nature (good and bad, light and darkness, wet and dry, and so forth), there is also a triad of moderation that adds depth to reality. These three dimensions define reality: past, present and future; liquid, solid and gas; width, length and height. It is the latter of these that is of interest to us right now.
Without a third point, no shapes can be formed. This is true in plane geometry as well as solid geometry. It is in two dimensions, at a time, that basic shapes are made on a flat surface. On that surface, a triangle is the most stable of all shapes. This is because once the three sides are connected, the angles are fixed in place. To change an angle, one of the sides has to be compromised. The changing of the length of a side changes the angles as the sides are reconnected.
Going back to the right triangle, the one with the set ratio of 3:4:5, let us say that side 'a' (3) is shortened to 2 units (let's use meters). To maintain the right triangle, the side opposite the right angle now must be shortened as well. The new length will be the square root of 20 (a little less than 4.5 meters). To save the 5 meter pole, all the angles change as well. Rotating outward to about 108 degrees, the 5 foot side once again secures to an stable, though offset triangle. At a ratio of 2:4:5, the ratio is definitely not "right."
And now, let me show how the sum of the three angles will always be twice that of a right angle. The reason I don't give this in degrees is because the presently defined circle of 360 degrees is based on assuming the "base 60" system of antiquity. It makes dividing a circle into even numbers quite easy, but is not "known" in all cultures. For the record, 360 is 12 times 30, or 3 x 4 x 5 x 6.
So, a right angle was formed halfway along an arch which will make a straight line. If you take the original length of cord - that is 3 meters - you can make a semi-circle defined by a straight line. This forms the base for two identical triangles, each with the ratio of 3:4:5. Let's take the cord that forms the original triangle and move the second peg (point b) that was three meters out. Moving it further from point c, the new position of the peg increases the angle of angle BCA while reducing the other angles. All along, the length of the cord remains twelve (3 + 4 + 5) meters long. The area inside the cord diminishes toward 0 until the two halves of the cord reach a length of 6 meters.
The cord is now twice the length of the perpendicular line that formed the right angle and the angle of that line is also twice that of the right angle. At all times, the angles inside the changing triangles added up to twice the right angle. If the peg had move toward point c, at some point the sides would have become equal, with each angle also being equal, or one third the angle of a straight line. In this case, 3 sides of 4 meters each.
So, What do I know?
A triangle is defined as a shape on a plane which has three lines intersecting at three angles.
The angles within any triangle will add to that of a straight line. This total will be twice the angle formed by a perpendicular line bisecting that line.
In a triangle with three equal angles, the sides will be of equal length. The angles will add up to that of a straight line.
In mathematical terms:
Right Angle BAC = 1/2 x (Angle ABC + Angle BCA)
Triangle ABC = Angle BAC + Angle ABC + Angle BCA
Given Right Angle BAC = 90 degrees, line segment BC will have 180 degrees.
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