Triangles

I've been interested in mathematics ever since I discovered trigonometry. Trig is amazingly useful mathematics, not that "usefulness" is a good measure of intrinsic value. Some things are good regardless of their usefulness. Mountain climbing comes to mind. I don't think those guys who climbed Mt. Everest were practicing for the day they might need to climb a mountain. They just did it "because it was there". So it is with mathematics. Regardless, even the craziest math ever devised eventually found some practical use in the real world. One example is how studies of the fourth dimension helped Einstein form his Theory of Relativity. Another is how prime number theory led to cracking the German World War II secret code, and to the creation of new un-crackable codes.

Trigonometry means "triangle measurement" and trigonometry is the branch of mathematics that studies the properties of triangles. Now, you may ask, "What possible properties could something as simple as a triangle have?". It's a reasonable question. At first glance, there doesn't seem to be much to be said about them. And why would anyone be interested in triangles in the first place? Another reasonable question. I'll try to answer both.

Why Triangles?

Most inventions in mathematics were motivated by the practical needs of people. One of these is measuring the area of land for property deeds, determining value, and for planning building projects. It turns out that the area of many different complex shapes can more easily be found by decomposing the shape into non-overlapping triangles. The areas of the triangles are calculated, then added up. So knowing how to calculate the area of a triangle is very basic and useful.

Another important use for triangles derives from a unique property of triangles. Triangles are the only "rigid" shape made from simple straight lines. Imagine nailing three sticks together to form a triangle. The resulting structure is rigid. A square made from four sticks is, by comparison, quite flexible and weak. Because of their rigidity and strength, triangles are often used in construction materials and designs. For example, triangular-shaped steel constructions called "trusses" are used to build very strong bridges.

Another use for triangles is navigation and "direction-finding". Imagine trying to find your way around on the ocean, where there are no markers or street signs. Navigators always know exactly where they are by measuring the angles and sides of triangles. Direction-finding means locating the source of a disturbance such as an earthquake or an emergency radio signal. By measuring the angles to the disturbance from two different locations, the source location can easily be calculated using a method called "triangulation". Triangles also have theoretical importance to mathematicians because they form the basis of the trigonometric functions. Finally, they are important to physicists, engineers, and scientists because the trigonometric functions are able to describe an enormous range of natural phenomena.

I am often surprised by the sheer number of mathematical and geometric properties of something as "simple" as the humble triangle. Here are just a few of them.

Some Properties of Triangles.

I already mentioned the "rigidness" property above. Another property is that the area of any triangle equals one-half the height times the base. This formula for finding the area of a triangle can be derived from the diagram below.

Another property has to do with what geometers call "central measures" of a triangle. I'm going to describe only two of the many central measures of a triangle. One of these central measures is called the centroid of the triangle, also known as the "center of gravity". Here's how to locate the centroid of any triangle. From each vertex (corner point) of the triangle, draw a line (Median) to the midpoint of the opposite side. The three lines will intersect in a single point inside the triangle. This point is the centroid.

The centroid has physical, as well as geometric significance. It is the point where the triangle will balance if cut out of some uniform material. Look at one of the medians and at the position of the centroid on that median. You'll see that the centroid is closer to the triangle's side than the vertex. This is true for all three medians. In fact, it can be proven that the centroid divides each median into two segments in the ratio of exactly 2:1. Another surprising fact is that each median divides the triangle into two triangles of equal area. Because of this, each of the six little triangles also have equal area. Just looking at the different lengths, angles, and shapes of the triangles, I don't think I would have guessed any of this. Triangles are deep.

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The incenter is another type of central measure. It is the point inside the triangle that is the same distance to all three sides. Imagine you owned a piece of land shaped like triangle ABC in Figure A. below. The sides of the triangle are roads bordering your land. The dotted lines represent driveways that lead from your house (black dot) to each of the three roads. Where would you build your house so that you had to drive the same distance to any of the three roads? In other words, where should you locate the black dot so that the three dotted lines are of equal length? Figure A shows the wrong place to build the house. ( The dotted distances to the sides are perpendicular [i.e. shortest] distances to the sides. - My drawing is a little off)

(A different but related question is: Where should you build your house to minimize the total distance to all three roads? - I'll take that problem up in a later update.)

We're going locate the incenter using the "classical geometric tools", the compass and straightedge. These are the basic tools that were used by the ancient Greek geometers. These two tools were chosen because they allow only the basic maneuvers of drawing a circle, drawing a straight line, and replicating a distance. No measurement is allowed and there are no markings on the straightedge.

Using only the compass and straightedge, certain geometric "constructions" are possible. For example, given an arbitrary distance (a straight line segment), it is possible to make a pencil mark that divides that distance exactly in half. This is done without any measurement and with no estimation. Now, it is to be understood that since we are doing this on paper with imperfect tools, by exact, I mean theoretically exact. Geometric constructions are not meant to be construed as exact in a practical sense, but rather in a logical sense. Another basic construction is to divide (bisect) an arbitrary angle into two equal angles. This is the construction needed to find the incenter.

If you don't remember these from high school geometry class, here's a review: Basic geometric constructions. Geometric constructions are a deep and interesting topic in it's own right. They are easy and fun to do and you can make some very beautiful and colorful designs using only a cheap compass and a ruler. It's understood that the markings on the ruler are not to be used for measuring. You won't need to. The edge of the ruler is used only to draw straight lines.

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The only construction needed to locate the incenter is to bisect an arbitrary angle into two equal angles. In Figure B below, the dotted line bisects angle A. Now, any point on the dotted line is the same distance from AB as it is from AC. You should convince yourself that this is true before going any further. So, the incenter must be somewhere on the dotted line.

Now we draw another bisector through, say, angle C as in Figure C below.

By the same logic, the incenter must also be somewhere on this dotted line. Therefore, the incenter must be at the point of intersection of the two lines. As a check on accuracy, you can construct the third bisector through angle B. All three bisectors should intersect at the same point - the incenter. Once the incenter is located, you can construct perpendicular lines to the three sides as shown in Figure D below. The three perpendicular lines will all have equal length.

Since all three lines have equal length, they can be the radii of a circle with the black dot as the center. As shown in Figure E. below, the "inscribed" circle in the triangle just touches (is tangent to) the three sides of the triangle. Each perpendicular line is a radius of the inscribed circle.

Next update I'll show how it's also possible to "circumscribe" a circle around the whole triangle so that the points A, B, and C all lie on the circle. For now, see if you can figure out how to do it. Here's a hint: You have to find the center of this circle - it's the point that's the same distance from A, B, and C, so first find the points that are the same distance to A and B.

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