## Integer Triangles such the median & Altitude to/from the largest side are all integers

To find triangles   $(a, \; b, \; c)$   such that its sides, the median   (   $m_c$   )   to the largest side   $c$
and the altitude from   $c$   are all integers.

In   $(a, \; b, \; c)$   triangle, the lengths of the medians :
$(2 \, m_a)^2 \; = \; 2 \, b^2 \; + \; 2 \, c^2 \; - \; a^2$
$(2 \, m_b)^2 \; = \; 2 \, c^2 \; + \; 2 \, a^2 \; - \; b^2$
$(2 \, m_c)^2 \; = \; 2 \, a^2 \; + \; 2 \, b^2 \; - \; c^2$

If   $A$   is the area of a triangle whose sides have lengths   $a, \; b$,   and   $c$   then

$4 \, A \; = \; \sqrt { \,(a+b+c) \,(a+b-c) \,(a-b+c) \,(-a+b+c) \,}$

The altitude from side   $c$   is given by    $h_c \; = \; 2 \; \times \; A \,/ \,c$