### Random Post

### Recent Posts

### Recent Comments

Jim Randell on Enigma 1691: Factory part… | |

Jim Randell on Puzzle 52: Football on the Isl… | |

geoffrounce on Enigma 1691: Factory part… | |

Hugh Casement on Enigma 1070: Time to work | |

Jim Randell on Enigma 1070: Time to work |

### Archives

### Categories

- article (11)
- enigma (1,157)
- misc (2)
- project euler (2)
- puzzle (40)
- site news (44)
- tantalizer (42)
- teaser (3)

### Site Stats

- 177,808 hits

Advertisements

We have already solved a similar problem to this in

Enigma 1313(although it was set 19 years after this problem).The ratio of the area of the triangles

BQTandABTis:in our case:

We’re interested in the root where

0 < k < ½.So:

Which we can plug back into the equation we derived in

Enigma 1313for the ratio of the area of the central triangle to the overall triangle (or we can note that the problem text forEnigma 1313tells us that in the case ofk=2/5the ratio isR=1/19).So if the area of the central triangle is

X:Solution:The area of the fieldSTUis 5 acres.Here’s a Python program that uses SymPy to do the maths from scratch:

The total area of the farm is 95 acres, which is 19/128 of a square mile. So the farm is approximately 0.5855 miles along each edge, which is about 3091.4 ft (≈ 942.26 m).

Should it ever be needed in future the program makes the following determinations:

Each of the 3 corner fields (like BQT) accounts for

k³ / (1 − k + k²)of the total area of the triangle (in this case 8 / 95).Each of the 3 edge fields (like ASUR) accounts for

(k − k² − k³) / (1 − k + k²)of the total area of the triangle (in this case 22 / 95).The central field (STU) accounts for

(1 − 4k + 4k²) / (1 − k + k²)of the total area of the triangle (in this case 5 / 95 = 1 / 19).Link:[ https://enigmaticcode.files.wordpress.com/2018/01/rouths-theorem1.pdf ].Here is a more detailed determination of the various areas made when a triangle is split by 3 cevians [ see: https://en.wikipedia.org/wiki/Cevian ]. Results are given for when the cevians split the opposite sides by ratios or by fractional amounts.

The results for the ratio of the area of triangle XYZ to the area of ABC is known as Routh’s Theorem [ https://en.wikipedia.org/wiki/Routh%27s_theorem ].