Teaser 2865: Seventh heaven?
From The Sunday Times, 20th August 2017 [link]
I have a modern painting by the surrealist artist Doolali. It is called “Seventh Heaven” and it consists of a triangle with green sides and a red spot on each of its sides. The red spots are one seventh of the way along each side as you pass clockwise around the triangle. Then each of the red spots is joined by a straight blue line to the opposite corner of the triangle. These three blue lines create a new triangle within the original one and the new triangle has area 100 sq cm.
What is the area of the green triangle?
[teaser2865]
S2T2 
Jim Randell 10:43 am on 12 May 2020 Permalink |
We have encountered similar problems to this before (see: Enigma 1313, Enigma 320, Enigma 1076).
The generalisation of this puzzle is known as Routh’s Theorem [ @wikipedia ], which states that the ratio of the area of the central triangle to the original triangle, where the sides are divided in the ratios 1:x, 1:y, 1:z is given by the formula:
I made a note giving the areas of the various subdivisions here [ rouths-theorem.pdf ].
For this puzzle we have: x = y = z = 6, which gives the ratio as:
The smaller triangle has an area of 100, so the larger triangle has an area of: 100 × 43/25 = 172.
Solution: The area of the green triangle is 172 sq cm.
Or using a program:
from enigma import rational, printf # ratio ABC/XYZ in Routh's theorem def routh(x, y, z): a = x * y * z - 1 b = (x * y + y + 1) * (y * z + z + 1) * (z * x + x + 1) return rational(a * a, b) # compute the ratio of the area of the inner and outer triangles R = routh(6, 6, 6) # output the solution, given the inner triangle has area 100 printf("outer triangle = {x} [R={R}]", x=100 / R)LikeLike