# Sunday Times Teaser 2865 – Seventh Heaven?

*by RBJT Allenby*

#### Published August 20 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 cm².

What is the area of the green triangle?

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Since the result does not depend on the shape of the triangle, we are free to choose this to make the maths easy. The following diagram shows the triangle that I used with the (x, y) coordinates of the triangle’s vertices and the points on each of its sides:

The equations for the two lines that cross at point P are:\[x=14-13y\] \[x=(4/3)y\] Solving these equations to find the y value at point P gives:\[y=42/43\] Since the triangles ABC and BCP have the same baseline, the ratio of their areas is equal to the ratio of their heights, which hence gives:\[\frac{\Delta BCP}{\Delta ABC}=(42/43)/7 = 6/43\] The triangle ABC is made up of the central triangle and three triangles each with \(6/43\) of its area, which means that the central triangle has an area of \(25/43\) of its area. And knowing that the area of the central triangle is \(100 cm^2\) allows us to derive the area of triangle ABC as \(172 cm^2\).

Routh’s theorem deals with the more general case in which the sides are divided by the cevians in different proportions.

Hereis a short note deriving the area relationships for the general case.