All the centers of a triangle

Geometry Level 3

Imagine a triangle where all the four centers-- incenter , centroid , circumcenter , orthocenter --are aligned, i. e. they all lie on the same line but no point is overlapping with any other.

Which of the following is definitely true about this triangle?

It must be isosceles It must be obtuse It must be equilateral It must be right-angled It can be any triangle

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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1 solution

Marta Reece
Jul 15, 2017

Isosceles triangle has a line of symmetry, so all the centers have to be on this line, and as shown in figure, it is easy to make them all distinct.

The triangle may not be equilateral because in that case all the centers would become one.

It could have an obtuse angle, but we cannot say that it definitely does. Again as demonstrated by the figure which does not.

Likewise it could be a right triangle, but need not be, so we cannot say that it is.

And finally it is not true of any triangle as the centers generally do not align.

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