Isosceles Triangle Containers of Triangles - 4

This post answers a specific question raised in this recent post :

The question: To find the least area isosceles triangle that contains a given convex polygon P, Will this work: 1)first find the least area general triangle that contains P and then 2) find the smallest area isosceles triangle that contains this least area general triangle?
Note: The same question can be asked for least perimeter isosceles triangle container of P with the same method proposed -perimeter simply replaces area. Also note that algorithms have already been found to find both smallest general triangle containers of convex polygons (above linked page has pointers) and also for least area isosceles triangle containers of general triangles.

The answer is negative. This picture illustrates an example:



The construction goes thus: Take a flat obtuse isosceles triangle ABC (AC = BC). Extend AC a little to D and join D to E on AB with segment DE crossing BC at F such that the area of triangle CDF is slightly less than that of triangle BFE.

Consider the grey quadrilateral AEFC as the convex region for which the least area isosceles triangle is to be found. The smallest area general triangle container containing AEFC is obviously not ABC and is not entirely contained within ABC either. However, it is also clear that ABC is the smallest area isosceles triangle that contains AEFC. Since ABC does not contain the whole of the smallest area general triangle containing AEFC, the method proposed above will fail.

Note 1: The above basic question and the proposed method can be restated with perimeter replacing area. It can be seen that if we find the least perimeter triangle container of AEFC, it will not be entirely inside ABC which will however be the least perimeter isosceles container of AEFC (note that CD and EB are quite small and almost equal in length). So, the basic method of first finding the general smallest triangle container and then finding the smallest isosceles container of this general container will fail for both area and perimeter cases.

Note 2: It can also be shown that the basic method fails to find the smallest area right triangle containing a convex polygon as well, as shown in this figure:

The construction is made with area marked '1' slightly less than area '2'. Then for the grey region as the input convex region, the red triangle will be the smallest area general triangle container and the black right triangle will be the smallest area right triangle container and the latter does not contain the former. The same construction should work as a counter to the method for min perimeter right triangle containers as well.

So we could conclude that the question of finding the min area (and min per) isosceles triangle containers of a convex polygon may well be different from finding the min area (min per) general triangle containers of the convex polygon.