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Photo: DER SPIEGEL
In math, crazy things happen sometimes.
So today: You draw the diameter in a circle with radius 1.
And a second chord that cuts the diameter at an angle of 45 degrees - see drawing above.
The diameter divides the chord into two parts with lengths x and y.
But no matter where exactly the chord and diameter meet: as long as the angle between them is 45 degrees, the
sum x2 + y2
does not change.
Why not?
How big is this sum?
The sum
x2 + y2
is 2. Always.
One possible solution uses the mid-perpendicular of the tendon AB.
It divides this into two pieces of equal length with the length (x + y) / 2 also runs through the center of the circle M - see the following drawing:
Two right-angled triangles have been created through the vertical center line: CDM and ACM.
The CDM triangle is even isosceles because the angles at points D and M are both 45 degrees.
The track CD has the length
(x + y) / 2 - y = (xy) / 2
This then also corresponds to the length of the line CM in the right-angled triangle ACM, because CD = CM.
With this we also know the lengths of all three sides of the right triangle ACM.
According to the Pythagorean theorem:
AC2 + CM2 = 12
We now substitute (x + y) / 2 for AC and (xy) / 2 for CM and get:
(x2 + 2xy + y2 + x2 - 2xy + y2) / 4 = 1
We multiply both sides by 4. The terms + 2xy and - 2xy cancel each other out:
2x2 + 2y2 = 4
x2 + y2 = 2
I discovered this puzzle in the Facebook group "Math problems and puzzles" of the Russian puzzle expert Konstantin Knop.
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