Take any three points in the universe, that are not on the same line, and they form a triangle.  We also have seen how these three points can generate the 2-D Euclidean space where the triangle lives.  Oh, the things we can do with a humble triangle.

We can start by labeling its parts.  The points are connected by the three line segments called sides. Let’s call the point that is farthest away from the other two points, P₁.  This is also the point that connects the two longest sides.  We can label the point on the other side of the longest side P₂, and the remaining point P₃.

It makes sense to put the label S₁ for the side opposite the point P₁, and the same for S₂, and S₃.  The side S₃ then connects the points P₁ and P₂, and is the longest side, simply because we chose P₁ and P₂ to be on the longest side.  And so the side S₂ is the second longest side, and S₁ the shortest side with the same reasoning.  In math language S₁ is less than or equal to S₂ and S₂ is less than or equal to S₃ : S₁ <= S₂ <= S₃.

We simply chose the labels so this is true. The way you choose to label things makes things easy to remember and deal with.  We can also choose to label the angle A₁ to be next to the point P₁, A₂ to be next to P₂, and A₃ to be next to P₃.

Angles are measured as percentages of the “piece of pie” inside the angle in proportion to the whole pie.  You can find the angular percentage of A₁ by taking the length of the circular arc C₁ (in the diagram) and dividing it by the circumference of the whole circle, 2*π*S₃.

The angle in degrees is found by multiplying this percentage, C₁/(2*π*S₃), by 360 degrees in the whole circle:

A₁ = 360 * C₁/(2*π*S₃).

Mathematicians chose 360 “degrees” in a full circle because it can be divided equally into 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360 pieces.

Another way to measure angles is to use “radians”, chosen so that 2*π radians are in a full circle. Thus in radians,

A₁ = 2*π * C₁/(2*π*S₃) = C₁/S₃

Which is a more natural measure for angles based on the arc lengths of a circle.

If you really want to learn geometry, trigonometry, or build virtual worlds, then understanding the concepts we are about to explore gives you great power.

Suppose, for example, you want to find a cartesian grid that will map where these points are in space, then choosing good reference points and directions can make all the difference.

If you choose P₁ to be the reference point (the origin), then choose the “x” direction to point directly along the S₃ side toward point P₂, and pick the perpendicular “y” direction so that P₃ is on the positive side.  Then you placed this universal triangle on a grid as shown on the diagram above.  P₃ is the only point that does not lie on an axis.  The distance of P₃ from the y axis is labeled “x” and the distance of P₃ from the x axis is labeled “y”, as shown in the diagram.

This breaks the triangle up into two rectangular regions, as shown by the dashed lines in the diagram.  The area of the left rectangle is x * y, and the area of the right one is (S₃-x) * y.  If you add these two areas together you get a total area of S₃ * y.  It is easy to see that the area of the triangle, by symmetry, is exactly half of the area of the two rectangles, so:

Area of triangle = ½ * S₃ * y.

We can easily see this from the diagram. The left rectangle is made with two exact copies of the x, y, S₂ triangle so the angle, A₁, is the same on the top and bottom half of the first rectangle.  The same goes for the A₂ angle in the (S₃-x), y, S₁ triangles on the right.  Adding the angles together on the top junction of the rectangle, it is easy to see that:

A₁ + A₂ + A₃ = 180 degrees (universally true for all triangles).

From the diagram and by the Pythagorean theorem we can also see that:

 S₂² = x² + y²

S₁² = (S₃-x)² + y²

       = S₃² – 2*S₃*x + x² + y²  (and x² + y² = S₂²)

       = S₃² + S₂² – 2*S₃*x    (known as the law of cosines)

And when S₁ and S₂ are the same length then:

S₂² = S₁²

S₂² = S₃² + S₂² – 2*S₃*x

Thus by simple algebra

S₃² = 2*S₃*x

x = ½ * S₃

And so we see that when x is halfway between P₁ and P₂, then the sides S₁ and S₂ must be equal in length. We can also see that if S₂ is greater than S₁ then x must be greater than ½ * S₃, on the right-hand side of the midpoint.

We now can see that there are some natural restrictions on where P₃ can be in the above diagram, since S₂ is greater than or equal to S₁, then x >= ½ * S₃, and since S₂ is shorter than S₃, P₃ must be inside a circle of radius S₃.  Thus, P₃ is restricted to be in the shaded region shown in the diagram.  If P₃ was outside this region, then S₂ would be longer than S₃ or shorter than S₁ and we would need to re-label the points and sides accordingly and redraw our grid, then P₃ would again be in the region.

We now can supply the Cartesian coordinates of the three points in this grid:

P₁: (0,0)
P₂: (S₃,0)
P₃: (x,y)

Note that this applies to all triangles in the universe, as long as we follow the labeling guidelines for the parts of the triangle and set up the grid like we have done here.

To work with the angles and find x and y, there are “trigonometric” functions on our calculators, Sin(Angle) and Cos(Angle), to find x and y from the angles, that are defined as follows:

Sin(A₁) = y/S₂  [or y = S₂ * Sin(A₁)]

Cos(A₁) = x/S₂ [or x = S₂ * Cos(A₁)]

We can also work with the angle A₂ by looking at things from the P₂ side:

Sin(A₂) = y/S₁           [or y = S₁ * Sin(A₂)]

Cos(A₂) = (S₃-x)/S₁ [or x = S₃ – S₁ * Cos(A₂)]

The angle A₃ can be found because A₁ + A₂ + A₃ = 180 degrees.

Notice that now you can find x and y if you know the angle A₁ or A₂, and the sides.  You can fool around with this to find the missing information.  For example, combining both expressions for x and y shown above, we get:

   x = S₂ * Cos(A₁) = S₃ – S₁ * Cos(A₂)

   y = S₂ * Sin(A₁) = S₁ * Sin(A₂)    (known as the law of sines)

These relations can be used to find x and y from the angles and sides, or the angles from x and y, or the sides from combinations of each.  These are very useful in geometry.  You can explore all the possibilities by taking a trigonometry course.

A surprising and revealing fact: you can use reflection symmetry to find a point that is the same distance away from all three points P₁, P₂, and P₃.  We call this the “center” point, C, and thus you can draw a circle that connects all three points. This means that if you randomly choose any 3 non-linear points, you can find a unique circle that connects these points.

To show this, we start by recalling the wonders of reflection symmetry.  The midpoint of S₃ between P₁ and P₂ is the same distance from both these end points. We can find all the points that are the same distance away from P₁ and P₂ by drawing a perpendicular line of reflection through the midpoint of S₃.  We saw that this is the line where x = ½ * S₃.  Any point on this vertical line is the same distance away from P₁ as it is from P₂.  This is shown by the vertical dashed line in the diagram above.  In the same fashion, the points that are the same distance from P₁ as they are from P₃ are found to be along the line of reflection that cuts the segment S₂ in half, as is shown by the other dashed line on the diagram.  These two midpoint lines of reflection from S₃ and S₂ meet at the “center point” C.  And so, we see that this point C is the same distance “r” from P₁, P₂, and P₃.  We can draw a circle with center C and radius r that connects all three of these points, as shown in the diagram above.

Before we finish, let’s explore one other surprising universal fact about triangles. This comes from reflection symmetry of bisected angles that we have not covered yet.  Any angle made from two lines can be cut in half, this forms a line of reflection symmetry.  For example, in the diagram above, there is a dotted line that divides the angle A₁ in half.

Through this line of reflection, you can reflect the line passing through P₁ and P₃ onto the line passing through P₁ and P₂. It might take a minute to see it. If I draw a circle with its center on this line, then the half-circle above the line can be reflected onto the half-circle below the line.  This means that if the circle touches the line on the top, then by symmetry, it must touch the line on the bottom at exactly the same distance from P₁.

Similarly, the point where the bisector lines of symmetry of A₁ and A₂ meet is the center of a circle that touches all three sides of the triangle.  If you connect the three points where this circle touches all three sides with line segments, then you have a smaller triangle that breaks the triangle into three triangles that have two sides of equal length.  The smaller triangle is inscribed inside a circle, and forms three other triangles with two equal sides (these are called isosceles triangles).

When you stare at the diagram, do you see the multiple symmetries here?  You can divide any triangle into 6 isosceles triangles.  Also note that the angle bisectors pass through the center of the inscribed circle and thus are perpendicular bisectors to the sides of the small triangle inside.  The distance from the center of the circle to the sides is equal to the radius of the circle.  Believe me, understanding this makes geometry and trigonometry so much more simple.  May the wonders of symmetry never cease!