A Geometric Problem

I found the following problem on my Google+ account: In the image below, the total length of the blue line segments is equal to the total length of the green line segments. The segments form an angle of 60° among each other.

After a while of thinking, I found a nice proof of this phenomenon. It is probably the standard proof, and it also yields a lot of generalizations.

We assume the black dot at (0,0) in the plane and denote the direction of the green line segments by vectors by v1, v2, and v3, all of them with length 1. Then

$$v_1+v_2+v_3=0$$

If we denote the lengths of the green segments with a1, a2 and a3, we have that

$$a_1v_1, \quad a_2v_2, \quad a_3a_3$$

are the endpoints of the segments. Now, let c be the center of the circle. We thus have

$$\|c-a_kv_k\| = \|c+b_kv_k\|$$

for k=1,2,3 if we denote the lengths of the blue segments with b1, b2, b3. Squaring this and using a bit of vector algebra, we get

$$– a_k (c \cdot v_k) + a_k^2 = b_k (c \cdot v_k) + b_k^2.$$

Here, the dot denotes the scalar product of two vectors. Thus

$$a_k – b_k = c \cdot v_k$$

Summing up for k=1,2,3 we get our conclusion

$$a_1+a_2+a_3 = b_1 + b_2 + b_3.$$

This generalizes to finitely many vectors which sum up to 0. E.g., we can take the sides of the following pentagram as vectors.

The corresponding result can be seen in the next figure.

Of course, a special case is 10 line segments with equal angle 36° between them.

31. Dezember 2017 von mga010
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