Maximum number of triangles formed in a pentagon with equal area

Question

All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and 10 triangles. Find the maximum number of triangles with the same area that may exist in the division.

The best I could do (or anyone can do) is that to draw a regular pentagon, which will give 2 sets of triangles, each having equal area, which means the maximum number for now is 5.

Can it be done better? Also can it be proved that the answer obtained (by someone) will be maximum?

Paul Panzer · Accepted Answer

We can do

Things we can't do:

This still leaves open the possibility of

Jaap Scherphuis · Answer

This may be a bit too simple. and not the intended solution, but

AxiomaticSystem · Answer

Continuing Paul's result:

Let's check them, using the fact that since only areas are considered we can do arbitrary linear maps to fix points:

Therefore,

Maximum number of triangles formed in a pentagon with equal area

3 Answers

Add your own answers!

Ask a Question