Solution to "The Village Surveyors"

The solution to this problem depends on the properties of the affine transformation. This transformation has the form

X = a*x + b*y + c
Y = d*x +e*y + f

It transforms straight lines into straight lines and parallel lines into parallel lines and one of its properties is that the ratios of areas of figures remains the same after transformation. It has 6 parameters, which means that any triangle can be transformed into any other triangle.

In this problem, the layout of villages and trees is described solely in terms of straight lines and the points where they cross and these properties are unchanged in an affine transformation. Hence the ratios of areas of triangles in the layout will be the same for all figures satisfying this layout. Hence, if we can find a single pattern which obeys these rules and work out the ratios of its areas, then these ratios will be the same for all patterns.

We choose our single instance as follows:-

Draw a triangle whose sides are the three areas specified, 6 units, 8 units and 10 units. This will be a right angled triangle and its area will be 24 square units. Place the Oak , Ash and Cypress at its vertices (O A and C) and place the beech(B) at its incentre. (The incentre is the centre of the circle which touches all three sides.) Call the radius of the incircle R. The three triangles OBA, ABC and CBO will all have the same height R. Hence their areas will be 3*R, 4*R and 5*R. The total of these areas is the area of the triangle OAC which is 24 units so that R = 2. This gives us the first element of the problem, that is a triangle divided into 3 parts whose areas are 6,8,and 10 square units.

We now have to place the villages.

Draw a line through A perpendicular to BA and let it meet the extended line CB at Amwell (Am) and the extended line OB at Crofton (Cr).
Extend the lines through Am and O and Cr and C to meet at Boxford (Bo).

Simple geometry will then show that the lines through A and Bo, O and Cr, and C and Am are the altitudes of the triangle Am Bo Cr and meet at B.

Simple trigonometry will then show that the area of Am Bo Cr is 120 square units, so that the answer to the problem is 120 acres.

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