Golden triangle (geometry)

Golden triangles of the first type ( and ) and second type ( ); the red angle is each ${\displaystyle \Delta ABC}$ ${\displaystyle \Delta BCX}$ ${\displaystyle \Delta ACX}$ ${\displaystyle 36^{\circ }}$

In geometry and elementary geometry , a golden triangle is an isosceles triangle in which the lengths of the base and legs are in the ratio of the golden ratio . A distinction is made between the golden triangle of the first kind and the golden triangle of the second kind : The golden triangle of the first kind is an isosceles -acute-angled triangle and has the angles , and . The golden triangle of the second kind is an isosceles- obtuse triangle and has the angles , and . ${\displaystyle 72^{\circ }}$ ${\displaystyle 72^{\circ }}$ ${\displaystyle 36^{\circ }}$ ${\displaystyle 36^{\circ }}$ ${\displaystyle 36^{\circ }}$ ${\displaystyle 108^{\circ }}$

Determination of the angles

Elementary geometric

On the longest side of , you remove the shortest side , if necessary starting from the corner point with the smaller angle, and connect the resulting removal point with the opposite corner point. In this way it is split into two sub-triangles and . ${\displaystyle \Delta }$ ${\displaystyle \Delta }$ ${\displaystyle {\Delta }_{1}}$ ${\displaystyle {\Delta }_{2}}$

With the similarity theorems it follows that either or is too similar . From this one draws the conclusion that the sum of the interior angles is equal to five times the smallest angle. Hence, one of the angles is the same . If this is the angle at the apex of , then it is a golden triangle of the first kind. If it is a base angle , then it is a golden triangle of the second kind. With the inside angle sum theorem it results that in the first case the inside angle triple must be the same , in the second case on the other hand comes only into question. ${\displaystyle {\Delta }_{1}}$ ${\displaystyle {\Delta }_{2}}$ ${\displaystyle \Delta }$ ${\displaystyle 36^{\circ }}$ ${\displaystyle \Delta }$ ${\displaystyle \Delta }$ ${\displaystyle \Delta }$ ${\displaystyle (72^{\circ },72^{\circ },36^{\circ })}$ ${\displaystyle (36^{\circ },36^{\circ },108^{\circ })}$

Trigonometric

Golden triangle of the first kind

If there is one with a base and legs and , this means for and  : ${\displaystyle \Delta =ABC}$ ${\displaystyle AB}$ ${\displaystyle AC}$ ${\displaystyle BC}$ ${\displaystyle a={\overline {AC}}={\overline {BC}}}$ ${\displaystyle c={\overline {AB}}}$

${\displaystyle {\frac {c}{a}}={\frac {{\sqrt {5}}-1}{2}}}$

If the base angle is at and the angle is at the tip of , one obtains ${\displaystyle \alpha }$ ${\displaystyle A}$ ${\displaystyle \gamma }$ ${\displaystyle C}$ ${\displaystyle \Delta }$

${\displaystyle \cos {\alpha }={\frac {c}{2\cdot a}}={\frac {{\sqrt {5}}-1}{4}}}$

and further

${\displaystyle \alpha =72^{\circ }}$

and finally with the inside angle sum theorem

${\displaystyle \gamma =36^{\circ }}$

Golden triangle of the second kind

With the same considerations as above, one obtains

${\displaystyle {\frac {a}{c}}={\frac {{\sqrt {5}}-1}{2}}}$

and further

${\displaystyle \cos {\alpha }={\frac {c}{2\cdot a}}={\frac {1}{2}}\cdot {\frac {c}{a}}={\frac {1}{2}}\cdot {\frac {2}{{\sqrt {5}}-1}}={\frac {1}{{\sqrt {5}}-1}}={\frac {{\sqrt {5}}+1}{4}}}$

and thus

${\displaystyle \alpha =36^{\circ }}$

and finally with the inside angle sum theorem

${\displaystyle \gamma =108^{\circ }}$

Rule of thumb

The golden triangles are exactly those isosceles triangles that contain an angle of . ${\displaystyle 36^{\circ }}$

construction

Golden triangles of the first and second kind, also is a golden triangle of the first kind, see animation ${\displaystyle \Delta ABC}$

Euclid of Alexandria described in his work The Elements a special isosceles triangle , now known as the Golden Triangle. This triangle is found again in his description for an equilateral and equiangular pentagon with a given circumference.

The starting point is any route that has to be divided according to the golden ratio. The so-called inner division is used for this . According to the above picture golden triangles of the first and second kind , the point of intersection and thus the two sections and To find the two golden triangles of the first and second kind, the point with its equal distances to the points and after connecting the Points and with the point the golden triangle of the first kind and the golden triangle of the second kind arise ${\displaystyle {\overline {AB}},}$ ${\displaystyle X}$ ${\displaystyle |AX|}$ ${\displaystyle |BX|.}$ ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle X.}$ ${\displaystyle A,X}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle XBC}$ ${\displaystyle AXC.}$

Occurrence

The above-described decomposition of into the partial triangles ${\displaystyle \Delta }$ and yields both forms of the golden triangle. Both forms always appear together. They result regularly in the construction with compasses and rulers of regular pentagon and regular decagon . The angles , and can therefore only be constructed with a compass and ruler. ${\displaystyle {\Delta }_{1}}$ ${\displaystyle {\Delta }_{2}}$ ${\displaystyle 36^{\circ }}$ ${\displaystyle 72^{\circ }}$ ${\displaystyle 108^{\circ }}$

literature

• Siegfried Krauter: Experience elementary geometry . Spektrum Akademischer Verlag , Munich 2005, ISBN 3-8274-1644-2 .
• Theophil Lambacher, Wilhelm Schweizer (Ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools. Geometry. Edition E. Part 2 . 13th edition. Ernst Klett Verlag , Stuttgart 1965.
• Mario Livio : The Golden Ratio. The Story of Phi, the World's Most Astonishing Number . Broadway Books, New York 2003, ISBN 0-7679-0816-3 .

Web links

Commons : Golden Triangle (Mathematics)  - collection of images, videos and audio files

Individual evidence

1. ↑ Krauter: p. 200
2. ↑ In English-language sources (cf. e.g. Livio: The Golden Ratio . P. 79 . ) Is meant by Golden Triangle alone Golden triangle of the first type , while for the Golden triangle of the second type the name Golden gnomon (of gnomon , ancient Greek γνώμων , equivalent to pointer to the sundial ) is common.
3. ↑ Lambacher-Schweizer: p. 165
4. ↑ Krauter: pp. 199-200
5. ↑ Lambacher-Schweizer: p. 165
6. ^ Johann Friedrich Lorenz : Euclid's elements, fifteen books . Ed .: In the publishing house of the bookstore des Waysenhauses. Hall 1781, S. 61 ff . ( Euclid's Elements, Fourth Book, The 10th Movement. , Page 61: Describing an isosceles triangle ... , Page 62: Let it be a straight line, AB ... [accessed December 18, 2016]).
7. ^ Johann Friedrich Lorenz : Euclid's elements, fifteen books . Ed .: In the publishing house of the bookstore des Waysenhauses. Hall 1781, S. 62 ff . ( Euclid's Elements, Fourth Book, The 11th Movement. , In a given circle, ABCDE, an equilateral and equiangular pentagon ... [accessed December 18, 2016]).
8. ↑ Livio: p. 79
9. ↑ Krauter: p. 201