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

HERON ALTITUDE

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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