The "middle" of a triangle, Method #3: The incenter

We come to another idea of where the center of a triangle is, a method called the incenter.  This is the point that is the center of the circle that is inscribed within the triangle, tangent to all three sides.

Once again, the point is defined as the meeting place for three lines. This time, the lines are the angle bisectors for the three angles. Like the centroid, the incenter must be in the interior of the triangle, which is not true for the circumcenter of an obtuse triangle.

Triangle classification and circumcenters

On Monday, we started the discussion of the "center" of a triangle, put in quotation marks because it is too vague. The centroid is the center of gravity and is always in the interior of the triangle, but the circumcenter, the center of the circle that passes through all three vertices, does not have to be in the interior. Here are the three possibilities.

We are going to classify triangles by the size of the largest angle.  Since the angles always add up to 180°, we can only have one angle that is 90° or greater, and it is possible to have all the angles less than 90°.  If they are all less than 90°, the triangle is classified as acute and the circumcenter will be in the interior of the triangle.

The next possibility is that the largest angle is exactly 90°, which we call a right angle, so we classify these triangles as right triangles.  With a right traingle. the circumcenter will always be at the midpoint of the longest side, usually called the hypotenuse.

The last possibility is called an obtuse triangle, one whose largest angle is more than 90°. In this case, the circumcenter will not be in the triangle's interior.

Notice that a triangle inscribed in a circle creates three arcs. The total measure of those arcs is 360°. It turns out the measure of each arc is twice the measure of the angle that creates the arc. For example, if an inscribed triangle had angles 45°, 35° and 100°, the three arcs would measure 90°, 70° and 200° respectively. The center of the circle would inside the arc of 200° but outside the interior of the triangle itself.

(Note: all these illustrations were nicked from a webpage created by Kristina Dunbar of UGA.)

Tomorrow: the incenter of the triangle.

The "middle" of a triangle, Method #2:The circumcenter

Yesterday, we discussed the centroid, the simplest way to measure the center of a triangle. For a triangle drawn as three vertices and the lines that connect them, draw the lines from each vertex to the midpoint of the opposite line. They will meet in a single point and that point is the centroid.  Another way to do it is if you have the coordinates of the three points, the centroid's x value is the average of the x values and likewise its y value is the average if the y values.

Here is a different way to get a center of a triangle, the point that is the same distance away from each vertex.  The geometric solution is to find the midpoint of each line segment of the triangle then draw the perpendicular bisector of the segment. The three perpendicular bisectors will all meet at a single point. The red circle in this picture is the only circle that goes through all three points A, B and C, and as this picture shows, it's not really necessary to do all three perpendicular bisectors, because the third will also pass through the point indicated by the green arrow.

If we had the coordinates for A, B and C, we would use them to find the midpoints and slopes of each line segments, call them mid1, mid2 and mid3 and slope1, slope2 and slope3.  It's possible that one of the slopes is zero, but it isn't possible to have two slopes equal to each other, because that would mean two lines are parallel, impossible if they are sides of a triangle.  Choose two lines with non-zero slopes and take the slope of the perpendicular, which equals -1/slope.  Without loss of generality, let's assume that lines 1 and 2 don't have a zero slope.  Then we just need to solve for x and y in the following pair of simultaneous linear equations.

y - y_from_mid1 = -1/slope1(x - x_from_mid1)
y - y_from_mid2 = -1/slope2(x - x_from_mid2)

It might look daunting, but the methods are fairly straightforward.

One "unusual" attribute of a circumcenter is that it doesn't have to be on the inside of the triangle. In fact, the rules for the position are based on the classification of the triangle.

Tomorrow, classifications and circumcenters.

The "middle" of a triangle, Method #1:The centroid

If we have a line segment, the midpoint is a simply defined thing, the point that cuts the segment into two equal parts. In this picture of the triangle ABC, the unlabeled red dots on each of the blue lines are the midpoints and the tick marks are place to indicate that each line segment has been cut in half.  The segment BC has two copies of a single tick mark |, the segment AC has double tick marks // and the segment AB has triple tick marks \\\.

When we talk about the center of a traingle, there are several different ways to discuss it. One of the easiest is the centroid, which is here labeled G.  The method used here is to draw a line from each vertex to the midpoint of the opposite side.  You only actually have to draw two, because the third line is promised to cross at the same place.

You will notice that the orange line segments cut the blue triangle into six parts.  Because G is the centroid, the areas of triangles AGB, BGC and AGC are all equal. More than that, each of the smaller six triangles has exactly one sixth of the area of ABC.

The method above has no coordinate system tied to it. Instead, it is planar geometry done in the classic Greek style, with diagrams that can be drawn using only a straightedge and a compass.

In the drawing here, the three vertices are given coordinates, specifically (1, 2), (3, 4) and (5, 0).  The coordinates make drawing the lines from vertices to midpoints unnecessary, because instead we can just take the average of the x coordinates and the average of the y coordinates as our corresponding x and y values.  In this case, (1+3+5)/3 = 9/3 = 3, and (2+4+0)/3 = 6/3 = 2, so the centroid is (3, 2), the point marked in red.

Tomorrow, we will look at the definition of the circumcenter, a point that is the same distance away from all three of the vertices.