**From New Scientist #2471, 30th October 2004**

Draw a triangle *ABC*. On the side *AB* mark the point *P* such that *AP*=(2/5)*AB*, on *BC* mark *Q* such that *BQ*=(2/5)*BC* and on *CA* mark *R* such that *CR*=(2/5)*CA*.

Draw the lines *AQ*, *BR* and *CP*. Call the point where *AQ* and *BR* cross *X*, the point where *BR* and *CP* cross *Y* and the point where *CP* and *AQ* cross *Z*.

If you did the appropriate calculations you would find that the area of triangle *XYZ* is 1/19 of the area of triangle *ABC*.

I went through the whole procedure above again, but, this time with each occurrence of the number 2/5 replaced by the number *k*, which is between 0 and 1/2. This time I found that the area of triangle *XYZ* was 1/37 of the area of triangle *ABC*.

What was the number *k*?

[enigma1313]

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This Python program uses the SymPy to do the maths. It runs in 444ms.

Solution:k = 3/7.Here’s a diagram that explains how the maths works:

I’ve used an equilateral triangle, but a similar approach will work with any triangle and give the same result.

The equation derived for the ratio of the areas is:

which, as can be seen in this graph, starts at R=1 when k=0, drops to R=0 at k=½ and then rises back to R=1 at k=1, which is how we would expect it to behave.

And here’s a constructive numerical solution in Python using the

find_value()function from theenigma.pylibrary. It uses a simple right angled triangle for the construction (although any triangle (A, B, C) could be specified in the calculation of R). After the numerical calculation it finds the closest rational approximation with a denominator of 1000 or less to give as the answer. It runs in 46ms.This code uses the ability to specify pairs as the formal arguments in function definitions, which Python 2.7.6 doesn’t seem to have a problem with, but it doesn’t work in Python 3.4.0, which is a shame as I thought it was rather a neat (and useful) trick. (PEP 3113 [ http://legacy.python.org/dev/peps/pep-3113/ ] explains their removal in Python 3, and how to work around it ).

This is a special case of

Routh’s Theorem[ https://en.wikipedia.org/wiki/Routh%27s_theorem ], which states that the ratio of the area of the central triangleXYZto the original triangleABC, where the sides are divided in the ratios1:x, 1:y, 1:zis given by the formula:In this case the ratios are all the same

x = y = z, which simplifies the formula to:And for this problem the ratio is equal to 1/37

So each side is split in the ratio 3:4, hence the value of

kwe are looking for is 3/7.A detailed determination of the various areas of the sub-divisions of the triangle is given in this document: [ https://enigmaticcode.files.wordpress.com/2018/01/rouths-theorem1.pdf ].

There is another special case – the one-seventh triangle, where the ratio of the two parts of one side is 2:1 in Routh’s formula, which I see is equivalent to k = 1/3 in your earlier formula for R.

If k = 1/3 in your formula for R, then the central area XYZ is 1/7 of the main area ABC.

In Routh’s formula , in this case, x = y = z = 2 (ie k = 1/3), so the ratio of the central area to the overall area is (2 * 2 * 2 – 1) ^2 / (4 + 2 + 1) ^ 3

ie 7^2 / 7^3 = 1/7

There’s a neat graphical demonstration that the area of the larger triangle is 7 times the area of the small triangle, by placing six reflections of the central triangle touching its edges and vertices and trimming off the overlaps to fill in the gaps. See [ https://web.archive.org/web/20090711014214im_/http://www.randi.org/images/02-09-01-trianglesolution2.gif ].

(I found this at [ https://web.archive.org/web/20060427055758/http://www.randi.org/jr/02-09-2001.html ] where James Randi was sent it by Martin Gardner who attributes it to Hugo Steinhaus).

The central triangle area being 1/7 of the main triangle area became known as Feynman’s Triangle, apparently proved over dinner once by Feynmann, the famous physicist.

See: http://journal.geometryexpressions.com/pdf/feinmansteiner.pdf

The paper examines other ratios of the sides of the triangle, including 1/4, 1/7 and 1/8, which gives a central triangle area of 1/2 of the main triangle area.

There is also an interesting example of construction of the vertices of the inner 1/7 triangle with integer coordinates:

https://math.stackexchange.com/questions/1131161/how-do-i-find-the-area-of-a-triangle-formed-by-cevians