Post Reply 
Little explorations with HP calculators (no Prime)
04-08-2017, 05:19 PM (This post was last modified: 04-08-2017 05:20 PM by Han.)
Post: #159
RE: Little explorations with the HP calculators
Great post, Gerson! For anyone who is curious about the tedious algebra using Heron's formula as shown by Gerson, here is a text output from Maple 17:

Code:
> restart:
> eq1:=a^2=b^2+c^2-2*b*c*cosA;
                           2                  2    2
                          a  = -2 b c cosA + b  + c 
> eq2:=b^2=a^2+c^2-2*a*c*cosB;
                           2                  2    2
                          b  = -2 a c cosB + a  + c 
> eq3:=c^2=a^2+b^2-2*a*b*cosC;
                           2                  2    2
                          c  = -2 a b cosC + a  + b 
> s1:=1/2*(a+b+c);
                               1     1     1  
                               - a + - b + - c
                               2     2     2  
> solve({eq1,eq2,eq3},{cosA,cosB,cosC});
      /          2    2    2          2    2    2          2    2    2\ 
      |         a  - b  - c          a  - b  + c          a  + b  - c | 
     < cosA = - ------------, cosB = ------------, cosC = ------------ >
      |            2 b c                2 a c                2 a b    | 
      \                                                               / 
> cosA := -(1/2)*(a^2-b^2-c^2)/(b*c); cosB := (1/2)*(a^2-b^2+c^2)/(a*c); cosC :=
> (1/2)*(a^2+b^2-c^2)/(a*b);
                                   2    2    2
                                  a  - b  - c 
                                - ------------
                                     2 b c    
                                  2    2    2
                                 a  - b  + c 
                                 ------------
                                    2 a c    
                                  2    2    2
                                 a  + b  - c 
                                 ------------
                                    2 a b    
> b:=2*r; c:=3*s; a:=4*t;
                                     2 r
                                     3 s
                                     4 t
> eq5:=u^2=s^2+(b-r)^2-2*s*(b-r)*cosA;
                           2     1  2   1  2   8  2
                          u  = - - s  + - r  + - t 
                                 2      3      3   
> eq6:=v^2=t^2+(c-s)^2-2*t*(c-s)*cosB;
                           2     5  2   5  2   2  2
                          v  = - - t  + - s  + - r 
                                 3      2      3   
> eq7:=w^2=r^2+(a-t)^2-2*r*(a-t)*cosC;
                           2     1  2      2   27  2
                          w  = - - r  + 3 t  + -- s 
                                 2             8    
> s2:=1/2*(u+v+w);
                               1     1     1  
                               - u + - v + - w
                               2     2     2  


> H1:=expand(s1*(s1-a)*(s1-b)*(s1-c));
                 2  2       2  2   9  2  2       4    4   81  4
              8 t  r  + 18 t  s  + - r  s  - 16 t  - r  - -- s 
                                   2                      16   
> H2:=expand(s2*(s2-u)*(s2-v)*(s2-w));
             1  2  2   1  2  2   1  2  2   1   4   1   4   1   4
             - u  v  + - u  w  + - v  w  - -- u  - -- v  - -- w 
             8         8         8         16      16      16   
> u2:=-1/2*s^2+1/3*r^2+8/3*t^2; v2:=-5/3*t^2+5/2*s^2+2/3*r^2;
> w2:=-(1/2)*r^2+3*t^2+(27/8)*s^2;
                               1  2   1  2   8  2
                             - - s  + - r  + - t 
                               2      3      3   
                               5  2   5  2   2  2
                             - - t  + - s  + - r 
                               3      2      3   
                              1  2      2   27  2
                            - - r  + 3 t  + -- s 
                              2             8    
> H2:=expand(-1/16*u2^2-1/16*v2^2-1/16*w2^2+1/8*u2*v2+1/8*u2*w2+1/8*v2*w2);
          49  2  2   49  2  2   49   2  2   49  4   49   4   441   4
          -- t  r  + -- t  s  + --- r  s  - -- t  - --- r  - ---- s 
          72         32         128         36      576      1024   
> simplify(H1/H2);
                                     576
                                     ---
                                     49 
>

Graph 3D | QPI | SolveSys
Find all posts by this user
Quote this message in a reply
Post Reply 


Messages In This Thread
RE: Little explorations with the HP calculators - Han - 04-08-2017 05:19 PM



User(s) browsing this thread: 2 Guest(s)