Here are some primitive Heron triangles for which

is a perfect square.

then, is also a Heron.

Can you find other such primitive Heron triangles?

(5,5,8) …………… isosceles obtuse triangle

(13,13,10) ………… isosceles acute triangle

(32,65,65) ………… isosceles acute triangle

(130,97,97) ……….. isosceles acute triangle

(72,85,85) ………… isosceles acute triangle

(170,157,157) ……… isosceles acute triangle

(72,325,325) ………. isosceles acute triangle

(650,397,397) ……… isosceles obtuse triangle

(128,1025,1025) ……. isosceles acute triangle

(2050,1153,1153) …… isosceles obtuse triangle

(200,629,629) ……… isosceles acute triangle

(1258,829,829) …….. isosceles obtuse triangle

(200,2501,2501) ……. isosceles acute triangle

(5002,2701,2701) …… isosceles obtuse triangle

(288,145,145) ……… isosceles obtuse triangle

(290,433,433) ……… isosceles acute triangle

(288,5185,5185) ……. isosceles acute triangle

(10370,5473,5473) ….. isosceles obtuse triangle

(392,2405,2405) ……. isosceles acute triangle

(4810,2797,2797) …… isosceles obtuse triangle

__Claim__ :

When the Heron is isosceles and is a square, then the perimeter of the triangle is a square number.

Can you explain why?

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Paul found examples of primitive Heron (non-isosceles) for which the perimeter of the triangle is not a square number:

(1)

area = 2520

perimeter = 378

area = 18270

perimeter = 756 =

(2)

area = 3060

perimeter = 510

area = 33150

perimeter = 1020 =

(3)

area = 11088

perimeter = 594

area = 64350

perimeter = 1188 =

There is another one that doesn’t have a square perimeter (last column). Format follows as above

{{5,5,8},12,18,60,{13,13,10},60,6}

{{29,174,175},2520,378,18270,{349,204,203},18270,6 Sqrt[21]}

{{32,65,65},1008,162,4680,{130,97,97},4680,18}

{{72,85,85},2772,242,11220,{170,157,157},11220,22}.

Paul.

a couple more the same

{{39,221,250},3060,510,33150,{471,289,260},33150,2 Sqrt[255]}

{{150,169,275},11088,594,64350,{444,425,319},64350,6 Sqrt[33]}

P.

Good catch! My statement is not accurate. It works for Heron isosceles

I’ve modified my claim.