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Photo: DER SPIEGEL
There are amazing phenomena in geometry.
As a child, I was fascinated by Thales' theorem.
Perhaps you remember: Every triangle, one side of which is the diameter of a circle and the third corner point of which is also on this circle, is right-angled.
The more general variant of this set is the peripheral angle set.
The following puzzle is also about a constant angle.
There are two equilateral triangles of equal size.
They touch at their lower corner points - see drawing above.
Two black lines connect the outer corner points with the upper corner point of the other triangle.
Show that the angle highlighted in yellow, at which the two lines intersect, is always the same, no matter how the two triangles are tilted towards one another!
How big is this angle?
The yellow angle has a size of 120 degrees.
The solution isn't too difficult if you take a closer look at the angles on the sketch below.
The two connections between the two triangles divide the 60 degree angle on the left and right outside into two angles.
In the following sketch these are colored pink and dark red at the triangle on the left.
The angles are together 60 degrees.
We also see that two edges of the two equilateral triangles and the right connecting line form an isosceles triangle.
In the sketch above, this triangle is highlighted with a darker shade.
With this we also know that the dark red angle appears again - namely at the top left next to the blue 60-degree angle.
Now we can take a closer look at the interior angles of the triangle at the top left, which is filled with lighter color.
One angle is pink, the second is made up of the blue 60-degree angle and a dark red-colored angle.
Because pink and dark red add up to 60 degrees, the third angle on the right must also be 60 degrees - only then will the sum of all three interior angles be 180 degrees.
This angle is therefore also marked in blue.
The yellow angle you are looking for is then simply the difference between 180 degrees and 60 degrees - i.e. 120 degrees.
We have also shown that the specific size of the red and pink angles does not matter.
It is only important that both together always result in 60 degrees.
I discovered this pretty geometry puzzle on the maths portal Mathigon.
If you missed a puzzle from the past few weeks, here are the most recent episodes:
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The fly in the glass
How big is the yellow rectangle?
The main thing is 1000
Heads or tails
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How big is the yellow triangle?
Which stick is magnetic?
Who will get the last coin?
Difficult legacy