Kurschak's Theorem, a Geometry Jewel.

József Kürschák (14 March 1864 – 26 March 1933) was a Hungarian mathematician noted for his work on trigonometry, but he may be more well known today, by the very few who seem to know of him, for a geometric gem that involves no Trig.  The basic idea is to prove , without trig, that the area of a regular dodecagon inscribed in a circle with unit radius is exactly three.  [a footnote about unusual things to share with students; if you inscribe a regular polygon in a unit circle, there are only two of them that will have a rational area.  I just gave away one of them, the other should not be too difficult to discern.]

The classical approach for any regular polygon is the one used by MathWorld.  

The area of the dodecagon (n=12) inscribed in a unit circle with R=1 is

 \(A=\frac{n}{2} R^2 sin(\frac{2\pi}{n})=3. \)

[Teachers may discuss among themselves the appropriate deduction for a correct result that insults the entire idea of the theorem...?]

So how did he do it?  Well the essence of the pretty image at the top may mask some of the deep ideas so let's part the haze a little.  This diagram shows all the elements of the solution.  

Equlateral triangles are constructed on the sides of a square inwardly. Their apexes form a square. Prove that the midpoints of the sides of the latter and the intersections of the side lines of the triangles form a dodecagon.

The inner square is the Kurschak Tile, and the source of a second name for the image.You can observe in the image that dodecagon is tiled by 24 isosceles 15°-15°-150° triangles and 12 regular triangles. Two isosceles triangles sharing the base combine into a rhombus with the side equal to that of regular triangles.

The area of the square outside the dodecagon is composed of 8 isosceles triangles identical to the ones forming the Rhombl in the interior of the dodecagon, and an additional 8 regular triangles congruent to the ones inside.  

And now we apply two easy images fromKürschak's Tile, G. L. Alexanderson, Kenneth Seydel; The Mathematical Gazette, Vol. 62, No. 421 (Oct., 1978), pp. 192-196 

We first realize that if the circle has a radius of one, thevequilateral triangles that formed the original construction, and the tile they formed, have edges of 2 units, Thus the whole tile has an area of four, made up of four unit squares.

In one of those unit squares there are nine triangles that are inside that square so we mark them to remove.  

 

Now we look at the parts of the dodecagon in the other three squares and note that there are some blank spaces outside the dodecagon.  And in an instant you recognize that all nine of the squares from the interior of the dodecagon will nicely fill each of the missing parts outside the dodecagon in these three squares, filling a space of three square units.  

And as the cool kids say, QED

Added notes and images from *Cut -the-knot.org , *  Wolfram MathWorld.  and