Supplementary exercise #6   Let triangle   ABC   and triangle   PQR   be as in Supplementary exercise #4 (from the previous assignment),

         
and let   (r,s,t)   be the orthocenter of triangle   ABC.  

1. Use the results of supplementary exercises 4 and 5 to find a formula for the barycentric coordinates of the circumcenter of triangle   ABC   (in terms of   r, s,   and   t).
   Suggestion: We're given that orthocenter of triangle   ABC   is   (r,s,t)^ABC   in barycentric coordinates. In rectangular coordinates, this is equivalent to saying that the orthocenter is   rA + sB + tC.   Of course, there's a similar equivalence in the case of triangle   PQR.   Making appropriate substitutions and doing some algebra, you can obtain an answer in rectangular coordinates and then convert it into barycentric coordinates.
2. Use the result of part a to show that the orthocenter, the circumcenter, and the centroid of triangle   ABC   are collinear.
    

Supplementary exercise #7   [Optional for extra credit]  

1. = Exercise 73 in §3.12 of the text.
   Suggestion: We can use Theorem 6 as one of our starting points, of course. In addition, we have the result of Supplementary Exercise #2, which we can combine with Threorem 6 to find the values of the ratios   ^r/_s ,   ^r/_t ,   and ^s/_t .   { More literally, if we set   ^r/_s ,   [for instance] equal to the value predicted by Theorem 6 and then clear denominators, we obtain a linear equation in the unknowns   r   and   s.   Proceeding in this way, we get a system of linear equations in the unknowns   r, s,   and t.
    
2. Use Theorem 7 [or Corollary 10] of Chapter 3 of the text, along with the result of Supplementary Exercise #7 to derive the formula for the Circumcenter given in Theorem 13 of Chapter 3 of the text.
   Suggestion: Presumably you've already done all of the really heavy lifting by now, and this part should follow by a fairly simple substitution.

Hint for problems #2 and #3 in §5.9:   To answer the question about orientation of an isometry obtained by composition or conjugation, we need to find out whether the determinant of its matrix part is +1 or -1. The main linear algebra fact needed for this purpose is that if   U   and   V   are   n by n   matrices, then the determinant of the product is equal to the product of the determinants; thus:
           det(UV) = det(U)·det(V).
And also, in the case of an invertible matrix  U,   we have:
           det(U ^-1) = ¹/_det(U) ,
because the identity matrix has determinant = 1 and   U·U ^-1 = 1   (the identity matrix).   Well, maybe one could claim that the formula for   det(U ^-1)   isn't such a big deal in the case where   U   is orthogonal, since we have   det(U) = ±1   in this case anyway. Nonetheless, it still could be the right way to think about things when you're considering conjugate matrices.
 

Note:   In addition to the problems already mentioned above, please remember that the assignment includes two other problems from §5.9 of the text, namely #4 and #12.