# Herons triangles with natural area, sides and at least two medians

Known eight primitives Herons triangles with natural area, sides and two medians, see A181928.

Let $$a,b,c$$ is half-sides, $$s$$ is area and $$m_a,m_b,m_c$$ is medians.

General properties such triangles:

$$s=(a + b + c) (b + c - a) (a + c - b) (a + b - c)$$ $$9s=(m_a + m_b + m_c) (m_b + m_c - m_a) (m_a + m_c - m_b) (m_a + m_b - m_c)$$ $$3\cdot5\cdot7\cdot8\mid s$$ $$m_a^2=2 (b^2 + c^2) - a^2$$ $$m_b^2=2 (c^2 + a^2) - b^2$$ $$m_c^2=2 (a^2 + b^2) - c^2$$ $$9a^2=2 (m_b^2 + m_c^2) - m_a^2$$ $$9b^2=2 (m_c^2 + m_a^2) - m_b^2$$ $$9c^2=2 (m_a^2 + m_b^2) - m_c^2$$ $$m_a^2 - 3 a^2 = 2 (3 b^2 - m_c^2) = 2 (3 c^2 - m_b^2) = x$$ $$m_b^2 - 3 b^2 = 2 (3 c^2 - m_a^2) = 2 (3 a^2 - m_c^2) = y$$ $$m_c^2 - 3 c^2 = 2 (3 a^2 - m_b^2) = 2 (3 b^2 - m_a^2) = z$$ $$x+y+z=0$$ $$m_a^2 + 3 a^2 = m_b^2 + 3 b^2 = m_c^2 + 3 c^2 = w$$ $$m_a^2 + m_b^2 + m_c^2 = 3 (a^2 + b^2 + c^2)=\frac{3}{2}w$$

where $$x,y,z$$ is some integers, $$w$$ is some positive integer.

Exact two half-sides is odd and one is even, exact two medians is odd and one is even.

$$w$$ is number of form $$(km \pm 3ln)^2+3(kn \mp lm)^2=(k^2+3l^2)(m^2+3n^2)$$, where $$k,l,m,n$$ is any positive integers.

Some relations between known primitives Herons triangles, selected by colors:

1) $$s=1680$$

$$a=26, b = 73, c = 51$$

$$m_a= 123.2233, m_b = 35, m_c = \color{red}{97}$$

$$x = 13156, y = -14762, z = 1606, w = 17212$$

2) $$s=221760$$

$$a = 626, b = 875, c = 291\vdots\color{red}{97}$$

$$m_a = 1144, m_b = \color{blue}{433}, m_c = 1493.4259$$

$$x = 133108, y = -2109386, z = 1976278\vdots\color{red}{97}, w = 2484364\vdots\color{red}{97}$$

3) $$s = 8168160$$

$$a = 3673, b = 4368, c = 1241$$

$$m_a = 5267.6447, m_b = 3314, m_c = 7975$$

$$x = -12724706, y = -46255676, z = 58980382, w = 68220868$$

4) $$s = 95726400$$

$$a = 13816, b = 28779, c = 15155\vdots\color{blue}{433}$$

$$m_a = 43874\vdots\color{magenta}{21937}, m_b = 3589\vdots\color{red}{97}, m_c = 42527.0663$$

$$x = 1352282308, y = -2471811602, z = 1119529294\vdots\color{blue}{433}, w = 2497573444\vdots\color{blue}{433}$$

5) $$s = 302793120$$

$$a = 11257, b = 14791, c = 14384$$

$$m_a = 26918.8823, m_b = 21177, m_c = 22002$$

$$x = 344466078, y = -207855714, z = -136610364, w = 1104786372$$

6) $$s = 569336866560$$

$$a = 1823675, b = 1930456\vdots\color{magenta}{21937}, c = 185629$$

$$m_a = 2048523\vdots\color{blue}{433}, m_b = 1730270.7575, m_c = 3751059\vdots\color{orange}{13\cdot96181}$$

$$x = -5780925035346, y = -8186144209212\vdots\color{magenta}{21937}, z = 13967069244558, w = 14173817998404\vdots\color{magenta}{21937}$$

7) $$s = 8548588738240320$$

$$a = 46263061\vdots\color{orange}{13\cdot96181}, b = 2442655864, c = 2396426547$$

$$m_a = 4839082088.5150, m_b = 2350198558\vdots\color{magenta}{21937}, m_c = 2488886435\vdots\color{#0a0}{661\cdot107581}$$

$$x = 23410294646947500526\vdots\color{orange}{13\cdot96181}, y = -12376269747775480124, z = -11034024899172020402, w = 23423136271826038852\vdots\color{orange}{13\cdot96181}$$

8) $$s = 17293367819066194215360$$

$$a = 31982445133, b = 356388643246, c = 336426334971\vdots\color{#0a0}{661\cdot107581}$$

$$m_a = 692364218455\vdots\color{orange}{13\cdot96181}, m_b = 318430912888, m_c = 378006355503.1853$$

$$x = 476299580606746896423958, y = -279640348821488939749004, z = -196659231785257956674954\vdots\color{#0a0}{661\cdot107581}, w = 482436841386859028750092\vdots\color{#0a0}{661\cdot107581}$$

Is it possible to use these relations to search next Herons triangles with natural two/three medians?