Heronian Perimeter

Geometry Level 5

Find perimeter of a Heronian triangle having integer sides and integer area for which Steiner inellipse has semi-major axis $a=319$ and semi-minor axis $b=\dfrac{124}{\sqrt{3}}$ .

The answer is 2178.

1 solution

Maria Kozlowska
Mar 4, 2016

The area of any triangle with that Steiner inellipse is $3\sqrt{3}ab=118668$ . Minimum side length is $248$ and maximum is $1105$ for two isosceles triangles. The only possible integer values are $965, 965, 248$ .

Hi; I wished you had mentioned that the Heronian triangle had integer sides, when I looked them up I found one site that said they could have rational sides with of course a rational area. I even found one such rational triangle with that inellipse and got the correct area, (that I can not understand at all). But I was so befuddled with the rational sides that I never saw the last step. Anyway, thanks for the problem.

bobbym none - 4 years, 10 months ago

I edited the problem. Sorry about the confusion.

Maria Kozlowska - 4 years, 10 months ago

Confusion, Cold Fusion, Confucius all are Greek to me. Problem was with me, not in your wording.

bobbym none - 4 years, 10 months ago

If you still have those rational triangle values, could you please post them? The area of a triangle to the area of its Steiner in-ellipse is constant. It is the same ratio as area of an equilateral triangle to the area of its in-circle. Any triangle can be thought of as a projection of equilateral triangle, and its Steiner in-ellipse is a projection of the original in-circle.

Maria Kozlowska - 4 years, 10 months ago

Hi; I do not have the work anymore, it took me half a century just to keep decent notes for problems I solved, I am afraid it will take me another half century to keep notes on those I do not. Probably was a hallucination anyway.

bobbym none - 4 years, 10 months ago

Heronian Perimeter

The answer is 2178.

1 solution

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