Prove that every Quadrilateral Area can be divided by cutting and joining into Four Identical Equal Shape/Area Pieces !

Did you really intend your challenge to apply to any class of quadrilateral?  There are many types.  For example, would one such as this be included?

Edited August 23, 20223 yr by OldChemE
clarification

Yes sure, it applies to this too !

Even this

 

3 hours ago, Commander said:

Yes sure, it applies to this too !

Even this

 

OK, show us how.

57 minutes ago, John Cuthber said:

OK, show us how.

..perhaps he is thinking of cutting the surface into infinitely small areas and combining them into one circle and dividing into four parts..

..wordplay.. He did not say how many cuts can be made..

 

Edited August 24, 20223 yr by Sensei

5 hours ago, Commander said:

Yes sure, it applies to this too !

Even this

 

How many sides does this figure have ?

 

As to the original question for a quadrilateral, any figure, even a disconnected one of several parts and/or curved lines, a line can be drawn the make the area on one side equal to the area on the other.

For some figures the two sides may be congruent figures or they may not depending upon the symmetries.

I do not know enough about the topology of quadrilaterals to answer this - it is certainly the sort of question Martin Gardner likes to ask and there is a five sided version in his book on this subject, but observe that if a dividing line can be srawn to from two congruent halves, also quads, then the process can be repeated to yield an answer.

 

It is only on quadrilaterals but as an odd one with obtuse angle was presented I did say that the question applies on that too and also realised it will be valid on the figure I gave though it looks like having more than 4 sides it is actually formed by 4 straight lines and thereby qualifies to be called a quadrilateral !

 

2 hours ago, Commander said:

It is only on quadrilaterals but as an odd one with obtuse angle was presented I did say that the question applies on that too and also realised it will be valid on the figure I gave though it looks like having more than 4 sides it is actually formed by 4 straight lines and thereby qualifies to be called a quadrilateral !

A small point;

 

Technically, the figure offered by OldChemE is a re-entrant quadrangle and the one you offered is a crossed quadrangle.

Neither is a quadrilateral.

A quadrangle is not a quadrilateral unless it is also a polygon.

 

Your figure might actually be a pair of triangles with a common vertex.

I can't tell from the sketch.

 

Edited August 24, 20223 yr by studiot

OK I will give the answer as one member has asked for it !

Any triangle can be split into four equal and identical parts each being similar triangle !

Any quadrilateral polygon can be divided into two triangles !! [even the odd figures]

Cutting these 2 triangles into 4 equal parts each enables you to join them into 4 pairs of identical shapes !!!

In the case of polygons with no crossing lines two of these shapes will occur naturally connected together each being similar to the parent polygon and the other two are obtained by moving and pairing the one-fourth triangles !!!!

That's it 🙂

8 hours ago, studiot said:

 

A small point;

 

Technically, the figure offered by OldChemE is a re-entrant quadrangle and the one you offered is a crossed quadrangle.

Neither is a quadrilateral.

A quadrangle is not a quadrilateral unless it is also a polygon.

 

Your figure might actually be a pair of triangles with a common vertex.

I can't tell from the sketch.

 

The figure I suggested is both a quadrilateral and a polygon: 

Concave quadrilaterals

In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside the quadrilateral.

polygon

/ (ˈpɒlɪˌɡɒn) /

 

---

noun

a closed plane figure bounded by three or more straight sides that meet in pairs in the same number of vertices, and do not intersect other than at these vertices.

 

I forgot to add, above. that in reading the original challenge I overlooked the word "joined."  With that word added I agree with Commander on the solution.

1 hour ago, OldChemE said:

I forgot to add, above. that in reading the original challenge I overlooked the word "joined."  With that word added I agree with Commander on the solution.

Thank you !

Members might find this distinction between a quadrangle and a quadrilateral enlightening.

It is from one of the many textbooks from that master of Euclidian geometry, C V Durrell, entitled Modern Geometry (from 100 years ago).

You might find the definitions in the first page useful.

Quadrangle deals with four Points and Quadrilateral deals with four line segments. All figures we discussed above satisfies these Conditions !

Whatever I have found and announced here that any such figure can be cut into four identical and similar figures seems to be a new finding though easy and simple !

On 8/24/2022 at 5:22 PM, studiot said:

 

A small point;

 

Technically, the figure offered by OldChemE is a re-entrant quadrangle and the one you offered is a crossed quadrangle.

Neither is a quadrilateral.

A quadrangle is not a quadrilateral unless it is also a polygon.

 

Your figure might actually be a pair of triangles with a common vertex.

I can't tell from the sketch.

 

Wiki disagrees.
https://en.wikipedia.org/wiki/Quadrilateral#/media/File:Six_Quadrilaterals.svg.

 

 

Illustrative Cuts

 

4 hours ago, Commander said:

Quadrangle deals with four Points and Quadrilateral deals with four line segments. All figures we discussed above satisfies these Conditions !

Whatever I have found and announced here that any such figure can be cut into four identical and similar figures seems to be a new finding though easy and simple !

Thank you for posting your solution to the question,   I am still thinking about it but +1.

Four line segments, yes not four lines.

Did you read the extract from Durrel  ?

You also mentioned area, do you include the diagonals as part of a tetragon ?

 

1 hour ago, John Cuthber said:

Wiki disagrees.
https://en.wikipedia.org/wiki/Quadrilateral#/media/File:Six_Quadrilaterals.svg.

 

 

Thank you for your contribution.

Can you explain what bearing it has on my comments ?

TY Studiot !

 

2 minutes ago, Commander said:

 

Illustrative Cuts

 

 

Thank you for this elaboration.

How are these of equal shape ?

 

It would be a very good idea if you could label your points (as in geometric points) so that issues can be discussed.

Well they are theoretically adding up to four equal shapes [some needs flipping and realigning] but it may appear not so as the drawings have not been done to scale ! But it gives the picture !!

Red = Blue = Yellow = Green

 

Edited August 26, 20223 yr by Commander