Hans Walser, [20130706]

Geometric Sequence and Triangles

Motivation: M. N. D., N.

Let *a, b, c* the sides of a triangle *ABC* such that (in this order)
is a geometric sequence. This means that we have a ratio *r* such that

.

Figure
1 gives an example with the ratio *r* =
1.2.

Fig. 1:
Ratio *r* = 1.2

We will see that the Golden Section plays an important role in special cases of these triangles.

In the
trivial case of *r* = 1 we have the
equilateral triangle (Fig. 2).

Fig. 2: Equilateral triangle

For *r* > 1 we
have *a* < *b* <* c*. The theorem of
Pythagoras yields:

The Golden Section comes in (Walser 2001 and Walser 2013). We get the large Right Golden Triangle of Figure 3.

Fig. 3: Large Right Golden Triangle

For *r* < 1 we have *a* > *b* >* c*. The theorem of Pythagoras yields in
this case:

We get the small Right Golden Triangle (Fig. 4).

Fig. 4: Small Right Golden Triangle

The small Golden Triangle fits into the large Golden Triangle (Fig. 5). It has the same shape like the large Right Golden Triangle, but a different orientation.

Fig. 5: Both Right Golden triangles

The
triangle inequality gives the condition *a*
+ *b* > *c*. Hence

On the
other side we have *a* < *b* + *c*.
That means

Therefore we have the range .

Note that the ratios of the small Right Golden Triangle, the equilateral triangle, and the large Right Golden Triangle are within this range:

Together with the range bounds they are a geometric sequence.

For the general triangle we let two of the three vertices of the triangle fix and study the locus of the third point. There are three cases:

(i)
*B* and *C* fix, *A* variable

(ii) *C* and *A* fix, *B* variable

(iii) *A* and *B* fix, *C* variable

In a Cartesian coordinate system we choose , , and (Fig. 6). For the sides of the triangle we get:

The geometric sequence condition is:

This
yields the implicit equation for the locus of *A*:

or

We have a curve of degree 4.

Fig. 6:
*B* and *C* fix

For x = 0 we find the equilateral triangle (Fig. 7).

Fig. 7: Equilateral triangle

Note
that the side *AB* of the equilateral
triangle is tangential to the curve. For the proof we use the gradient:

The
vector is orthogonal to
the curve, but also orthogonal to the side *AB*
of the equilateral triangle.

For we get the large Right Golden triangle (Fig. 8).

Fig. 8: Right Golden Triangle

We find also the small Right Golden Triangle (Fig. 9). Here we have

Fig. 9: Right Golden Triangle

Finally we find the Golden Section in different cases (Fig. 10 and 11). The major is indicated in blue and the minor in red.

Fig. 10: Golden Section

Fig. 11: Golden Section

Now we choose and (Fig. 12).

For the sides of the triangle we get:

The geometric sequence condition is again:

This
yields the implicit equation for the locus of *B*:

Fig.
12: *C* and *A* fix

The curve looks like an ellipse, but this cannot be, since it is a curve of degree 4. Nevertheless the difference to the ellipse with the same semi axes is very small (Fig. 13).

Fig. 13: Curve compared with ellipse

Again we find an equilateral triangle (Fig. 14), Right Golden triangles (Fig. 15) and the golden Section (Fig. 16).

Fig. 14: Equilateral triangle

Fig. 15: Right Golden Triangle

Fig. 16: Golden Section

Now we choose and (Fig. 17).

For the sides of the triangle we get:

The geometric sequence condition is again:

This
yields the implicit equation for the locus of *C*:

Fig.
17: *A* and *B* fix

The curve is symmetric to the curve of Figure 6.

Figure 18 shows the three curves in a triangular lattice. We see a lot of Golden Sections.

Fig. 18: The tree curves

References

Walser, Hans
(2001): *The Golden Section*.
Translated by Peter Hilton and Jean Pedersen. The
Mathematical Association of America 2001. ISBN 0-88385-534-8.

Walser, Hans (6. Auflage). (2013). *Der Goldene Schnitt*.
Mit einem Beitrag von Hans Wu§ing Ÿber populŠrwissenschaftliche
Mathematikliteratur aus
Leipzig. Leipzig: Edition am Gutenbergplatz. ISBN 978-3-937219-85-1.