Dividing a Quadrilateral into Equal Parts

[Albert1]

ABCD is a quadrilateral,and point P is a point on AD ,and

between points A and D

please construct a line (passing through point P),and

divide ABCD into two parts with equal area

 

[Evgeny.Makarov]

Re: quadrilateral

Do you know how Russian revolutionaries in the beginning of the 20th century wrote letters to their comrades from tsarist prisons? They would use milk to write hidden messages between the lines. (Yes, apparently, at that time, they served milk in prisons.) When milk was dry, it was invisible, but by holding the letter over a candle, the hidden message could be revealed. I have a feeling that something is written between the lines in this problem statement as well.

Spoiler

Edit: Apparently, the following is incorrect. There is no reason to believe that any line through the centroid cuts a quadrilateral into two parts with equal area.

Is it not the line that passes through P and the centroid? As Wikipedia helpfully reminds, the centroid can be found by constructing the centroid of triangles into which the quadrilateral is divided by diagonals.

 

[Opalg]

Re: quadrilateral

This strikes me as an extension of the http://www.mathhelpboards.com/f28/change-pentagon-into-triangle-equal-area-5486/.

[sp]Regard the quadrilateral ABCD as being a pentagon PABCD, with a $180^\circ$ angle at P. Then apply the construction in comment #6 in the pentagon thread. This gives a triangle with its apex at P, whose opposite edge (ST say) contains the edge BC of the quadrilateral, and which has the same area as the quadrilateral. Now bisect ST to get a point Q on BC. The line PQ will bisect the triangle and will therefore also bisect the quadrilateral.

I would include a diagram if I had time, but I expect Albert can provide one. (Smile)[/sp]

 

[Albert1]

Re: quadrilateral

This strikes me as an extension of the http://www.mathhelpboards.com/f28/change-pentagon-into-triangle-equal-area-5486/.

[sp]Regard the quadrilateral ABCD as being a pentagon PABCD, with a $180^\circ$ angle at P. Then apply the construction in comment #6 in the pentagon thread. This gives a triangle with its apex at P, whose opposite edge (ST say) contains the edge BC of the quadrilateral, and which has the same area as the quadrilateral. Now bisect ST to get a point Q on BC. The line PQ will bisect the triangle and will therefore also bisect the quadrilateral.

I would include a diagram if I had time, but I expect Albert can provide one. (Smile)[/sp]

yes the construction of the diagram is similar to the pentagon problem
now ! here is the diagram
https://www.physicsforums.com/attachments/1002._xfImport

 

[CellCoree]

I am not an expert in geometry or mathematics. However, based on my understanding of geometry, I can offer some insights on this topic.

Dividing a quadrilateral into equal parts can be achieved by constructing a line that passes through a specific point on one of its sides. In this case, the point P is on the side AD, between points A and D. To divide the quadrilateral ABCD into two equal parts, we need to construct a line that passes through point P and divides the area of the quadrilateral into two equal halves.

To do this, we can use the concept of perpendicular bisector. A perpendicular bisector is a line that divides a line segment into two equal parts, while also being perpendicular to that line segment. In this case, we can construct a perpendicular bisector of the line segment AD passing through point P. This line will intersect the side BC at a point, say Q. The line PQ will then divide the quadrilateral ABCD into two equal parts with equal areas.

To verify that the two parts have equal areas, we can use the formula for the area of a quadrilateral, which is ½ x base x height. Since the line PQ is a perpendicular bisector of AD, it will also bisect the height of the quadrilateral. Therefore, the area of the triangle APQ will be equal to the area of the triangle DPQ. Similarly, the area of the triangle CPQ will be equal to the area of the triangle BPQ. Therefore, the areas of the two parts of the quadrilateral ABCD will be equal.

In conclusion, by constructing a perpendicular bisector passing through point P on the side AD of the quadrilateral ABCD, we can divide it into two equal parts with equal areas. This method can also be used to divide a quadrilateral into any desired number of equal parts by constructing multiple perpendicular bisectors passing through different points on its sides.