Dividing an obtuse triangle into acute triangles

Can an obtuse triangle be subdivided into only acute triangles (right triangles are not allowed)?

Any number of subdivisions can be made as long as all of the angles in all resulting triangles are less than 90 degrees.

Example of an incorrect answer: Splitting a triangle with angles {120, 30, 30} by bisecting the obtuse angle results in two right triangles with angles {90, 60, 30}. This is incorrect because there are two angles which are not less than 90 degrees.


Experimentally it appears that seems that something close to Hagen von Eitzen’s comment of the construction using a regular pentagon

enter image description here

works for all obtuse triangles. Here is an example, with two isosceles triangles cutting off the sharp angles, and then choosing a suitable point inside the remaining pentagon to give 7 acute angled triangles in all. Not all pairs of isosceles triangles allow there to be a suitable interior point, but I could not find an obtuse angled triangle without some solution.

enter image description here

I would be surprised if there was a solution with fewer acute angled triangles.

Source : Link , Question Author : Joe C , Answer Author : Henry

Leave a Comment