Finding the Largest Triangle in a Polygon: Solutions Needed!

[damrai]

Hi,

I am a newmember and a newbie to this forum.

I am interested in Maths especially Geometry. I am trying to deal with a geometry problem of polygon and i need help.

My question is -

I have a polygon with n number of sides. I want to find out the area of the largest possible triangle from this polygon in two cases.

Case: 1 All of the vertices of the largest triangle found lie in the interior of the polygon, that is none of the points of the triangle are on the polygon.

Case: 2 All or any of the vertices of the triangle lie on the polygon

Any ideas, links , formulas, algorithms that can be helpful is highly appreciated

Thanks in advance.

Regards,

Damrai

 

[HallsofIvy]

If all of the vertices must be inside the polygon, there is no "largest" triangle.
Given any triangle with all vertices inside the polygon, you can move each vertex closer to the the polygon (say, half its distance from the polygon) and get a larger triangle.

 

[robphy]

I think some more information needs to be specified.

I assume that polygon is fixed (that is, it is not deformable) and, generally, not regular.

Is this polygon convex?

 

[damrai]

Hi,

Robphy and HallsofIvy thanks for your replies.

Assuming the polygon to be concave or convex. Also it is not a compulsion that all the three vertices lie within the boundary of the polygon.

HallsofIvy, you have mentioned that - "If all of the vertices must be inside the polygon, there is no "largest" triangle." - What if any two of the vertices lie on the polygon.


Awaiting for your replies.

Regards,
Damrai.

 

[koroljov]

If any two of the vertices lie on the polygon, you can still move the last vertex closer and closer to the polygon, thus there is still no largest triangle.

 

[HallsofIvy]

In other words, the "largest triangle" you can place in a polygon must have all three vertices on the polygon.

 

[damrai]

Hello,

Koroljov and HallsofIvy, can you please put down the mathematical representation for my problem.

I need some mathematical formula, postulates or any theorem which can help me to find out the area of the largest triangle from this polygon.

Many thanks in advance.

Regards,

Damrai.

 

[koroljov]

I think it can be solved this way (for a convex polygon):
Finding the triangle with the greatest area is the same as finding the triangle with the greatest circumference. (The formula of heroon: assume a triangle with sides a, b, c, and p=(a+b+c)/2, then the area=sqrt(p*(p-a)*(p-b)*(p-c))
Thus this might work:
If you remove a vertex of the polygon, the circumference of the polygon will change. You should remove the vertex of the polygon that causes the smallest change of circumference, and repeat this procedure until there are only 3 vertices left.

I'll post an image in some minutes.