Divide the lake equally

Geometry Level 5

There is a quadrilateral lake suitable for fishing. Four fishermen come and sit at each corner of the lake, the coordinates of which are $(1,1), (2,5), (4,4), (6,2).$ But they are in a conflict because there must be a fair chance. So, they've decided to find a point $(x, y)$ inside the lake such that the 4 line segments connecting the point and each of the midpoints of the quadrilateral divide the lake into 4 equal areas.

Help them find the point, and submit your answer as $x-y$ .

The answer is 0.5.

2 solutions

Niranjan Khanderia
Apr 23, 2017

Lines || to each diagonal through both vertices of the other diagonal would form a || ogram' PQRS.
The diagonal midpoint Z, where the diagonals of this || ogram intersect is the point required.

Lines through B(2,5) || A(1,1) and C(4,4) is y=x+3 , .......(1)
Lines through D(6,2) || A(1,1) and C(4,4) is y=x-4 , .......(2)

Lines through A(1,1) || B(2,5) and D(6,2) is y=- 3/4x+7/4 , .......(3)
Lines through C(4,4) || B(2,5) and D(6,2) is y=- 3/4x+7 , .......(4)

So one vertex Q of the ||gram from (1) and (3) is (-5/7,-5/7+3)=(- 5/7,16/7).
Opposite vertex S of the ||gram from (2) and (4) is (44/7,44/7-4)=(44/7,16/7).

So the midpoint of this diagonal QS, also point of intersection of diagonals, is the required point Z(39/14, 16/7).
So 39/14 - 16/7 = 7/14=1/2 .

Great solution sir, way faster than what I did! Still figuring out the reasoning behind. Until then this is nothing less than magic to me. :-)

Ujjwal Rane - 4 years, 1 month ago

Ajit Athle
Mar 18, 2017

Let the 4 given points be A:(1,1), B:(6,2),C:(4,4) & D:((2,5). Determine the midpoints of AB, BC, CD & DA. Let them be resply. P,Q R & S with Z:(x,y) as the required point within ABCD which gives us, Area (PZSA)= Area (QZPB)= Area (RZQC) =Area (SZRD). Now use the formula --- 2A=(x1y2 - x2y1)+(x2y3 - x3y2)+(x3y4 - x4y3)+(x4y1 - x1y4). Take care to go in the anti-clockwise direction around each quadrilateral so as to get +ve area in each case. Now two equations can easily be obtained to yield, x=39/14 & y=16/7 so that (x-y)=1/2. I apologize for not having learnt to use LaTeX yet.

There's an elegant method, given by Honsberger in Mathematical Gems III, to determine the location of Z. Draw lines through A & C parallel to the diagonal BD and lines through B & D parallel to diagonal AC. You now have a parallelogram the centre of which is the point "Z".

Ajit Athle - 4 years, 2 months ago

Lines through (2,5) || (1,1) and (4,4) is y=x+3 , .......(1)
Lines through (6,2) || (1,1) and (4,4) is y=x-4 , .......(2)

Lines through (1,1) || (2,5) and (6,2) is y=- 3/4x+7/4 , .......(3)
Lines through (4,4) || (2,5) and (6,2) is y=- 3/4x+7 , .......(4)

So one vertex of the ||gram from (1) and (3) is (-5/7,-5/7+3)=(- 5/7,16/7).
Opposite vertex of the ||gram from (2) and (4) is (44/7,44/7-4)=(44/7,16/7).

So required point is ((39/14, 16/7).
So 39/14 - 16/7 = 7/14=1/2 .
Mr. One Top, your help is very elegant and useful Thanks a lot..

Niranjan Khanderia - 4 years, 2 months ago

What is that equation?

Mas No - 4 years, 2 months ago

The equation above gives the area, A, of the quadrilateral formed by (x1,y1), (x2,y2), (x3,y3 and (x4,y4).

Ajit Athle - 4 years, 2 months ago

We could also find the centroid of the quadrilateral as there would be a symmetric distribution around it ( from definition of centroid ). Thus the point is $\frac{13}{2},6$ . Hence the answer.

Note : Take step by step centroid of triangles to prove $\frac{\sum_{i=1}^{4} (x_i,y_i}{4}$ .

Vishal Yadav - 4 years, 2 months ago

Divide the lake equally

The answer is 0.5.

2 solutions

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