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        riddles >> hard >> Dissecting a heronian triangle
        
(Message started by: JocK on Aug 18^th, 2005, 12:20pm)

Title: Dissecting a heronian triangle
Post by JocK on Aug 18^th, 2005, 12:20pm

Can you dissect a Heronian triangle into three Heronian triangles?

Title: Re: Dissecting a heronian triangle
Post by Barukh on Aug 18^th, 2005, 11:12pm

Yes.

Moreover, I think Heronian triangle can be dissected into any number of Heronian triangles.

JocK, do you want to strenghthen the requirements?

Title: Re: Dissecting a heronian triangle
Post by Grimbal on Aug 19^th, 2005, 1:41am

If you know a way to dissect a heronian triangle into 2 heronian triangles, it is trivial to dissect one into n heronian triangles.

Whatever a heronian triangle is... ::)

Title: Re: Dissecting a heronian triangle
Post by Grimbal on Aug 19^th, 2005, 4:12am

Funny. I reread the question and I see it asks to cut the triangle in 3. I would have sworn it said to cut it in 2, which explains my remark.

Title: Re: Dissecting a heronian triangle
Post by Grimbal on Aug 19^th, 2005, 4:20am

This should work:
If you have a triange ABC, where AB<AC, construct a similar A'B'C', mirror reversed and scaled down, such that
A' = A, C' = B, B' is on AC.
This cuts the triangle in 2. Repeat to have 3 pieces.
All sides should be rational, as are the surfaces.

It doesn't work for the equilateral triangle, but it isn't heronian anyway.

Title: Re: Dissecting a heronian triangle
Post by JocK on Aug 19^th, 2005, 1:24pm

on 08/18/05 at 23:12:04, Barukh wrote:

JocK, do you want to strenghthen the requirements?

Sure... ;D ... the follow-up question is:

Can you dissect a Heronian triangle in three Heronian triangles such that the resulting triangles have equal area?

(The much harder follow-up question I leave for later... ;) )

Title: Re: Dissecting a heronian triangle
Post by Barukh on Aug 22^nd, 2005, 9:54am

I don't know what is the much harder follow-up, but this equal-area variation is hard enough (at least for me).

The more I look at this problem, the more I think that the answer is "no", but cannot prove it. What I succeeded to show is that certain dissections (e.g. depicted below) are impossible.

Any comments?