The weight of a triangle

The hypotenuse of a triangle is the line that connects the vertex of the triangle to the midpoint of the opposite side. A triangle has exactly three lines of gravity and their intersection forms the triangle's centre of gravity.

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Look at the following triangle with the lines of gravity marked: The lines of gravity are marked in red. As with the heights of triangles, we denote the lines of gravity with a lower case letter t along with a subscript that specifies which side and vertex the line of gravity belongs to. Since opposite the vertex A we have the side a, the weight line will also be named t_a.

The hypotenuse halves a given triangle into two triangles with the same content. Unlike the height of a triangle, the weight lines look the same for all types of triangles.

The centre of a line segment

The three points S_a, S_b and S_c represent the midpoints of the given sides, of the given line segments. You can graphically calculate the midpoint using a compass and ruler. For example, you can find the centre of the side AB by drawing two equally sized circles centred at the vertices A and B. These circles must have a radius greater than half the length of the line segment AB - but this radius must be the same for both circles! Either estimate this, or draw a circle with a radius the length of the side AB. These two circles intersect at two points. Connect these points with a line and where this line intersects the side AB, there is the centre of the side AB.

The centre of gravity of the triangle, which is the imaginary centre of the triangle, is highlighted in green in the figure.

The centre of gravity

All lines of gravity always intersect at one point. This is how you know if you have drawn correctly. If they intersect somewhere off, you've either drawn inaccurately or completely wrong. The center of gravity is always inside the triangle, unlike the orthocenter, which can be outside the triangle.

The center of gravity is the imaginary center of the triangle, if you wanted to hold the triangle on the pencil points, then you should place the pencil just below the center of gravity to keep the triangle from falling off.

The centre of gravity is further divided by the lengths of the paperweights in a 1:2 ratio. This means that two-thirds of the length of the paperweight is on one side of the center of gravity and the remaining one-third is on the other side. The longer part of the weight line is always towards the "apex" of the triangle. "Near the side", on the other hand, is the shorter part. Look at the picture:

The length of the line segment AT (blue line) is twice the length of the line segment TS_a (green line). The segments AD, DT and TS_a are the same size. Similarly for the other heavy lines.

The middle transversal

The midpoint is the line segment that has the midpoints of the sides of the triangle as its extreme points. Examine the figure:

The points S_a, S_b and S_c are the midpoints of the sides, this remains the same as for the paperweight. To create the middle partition, we just connected the two points with a line segment each time. We denote the middle partitions by s along with the subscript where we put the opposite vertex. The triangle that results is called a transversal triangle. Interestingly, the center of gravity of this transversal triangle is the same as that of the original original triangle. In our case, it is the point T.

Calculating the length of the line of gravity

We can use Apollonius' theorem to calculate the length of the line of gravity. Assume now that a, b, c are the lengths of the corresponding sides of the triangle, similarly t_a, t_b and t_c are the lengths of the lines of gravity:

$$\begin{eqnarray} t_a&=&\sqrt{\frac{2b^2+2c^2-a^2}{4}}\\ t_b&=&\sqrt{\frac{2a^2+2c^2-b^2}{4}}\\ t_c&=&\sqrt{\frac{2a^2+2b^2-c^2}{4}} \end{eqnarray}$$