THEOREMS IN ELEMENTARY GEOMETRY



1)  Smarandache Concurrent Lines 
    ----------------------------
If a polygon with n sides (n >= 4) is circumscribed to a circle,
then there are at least three concurrent lines among the polygon's diagonals
and the lines which join tangential points of two non-adjacent sides.
 (This generalizes a geometric theorem of Newton.) 
                     
Reference:
  F. Smarandache, "Problemes avec and sans problemes!" 
  (French:  Problems with and without ... Problems!), Ed. Somipress,
    Fes, Morocco, 1983, Problem & Solution # 5.36, p. 54.




2)  Smarandache Cevians Theorem (I)
    -------------------------------
Let AA', BB', CC' be three concurrent cevians (lines) in the 
point P in the triangle ABC.  Then:

      PA/PA' + PB/PB' + PC/PC' >= 6,
  and
      PA     PB     PC     BA     CB     AC
     ---- . ---- . ---- = ---- . ---- . ---- >= 8.
      PA'    PB'    PC'    BA'    CB'    AC'

Reference:
  F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress,
    Fes, Morocco, 1983, Problems & Solutions # 5.37, p. 55, # 5.40, p. 58.




3)  Smarandache Orthic Theorem
    ---------------------------
Let AA', BB', CC' be the altitudes (heights) of the triangle ABC. 
Thus A'B'C' is the podaire triangle of the triangle ABC.  
Note AB = c, BC = a, CA = b, and A'B' = c', B'C' = a', C'A' = b'.  
Then:
      4(a'b' + b'c' + c'a') <= a^2 + b^2 + c^2

Reference:
  F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress,
    Fes, Morocco, 1983, Problem & Solution # 5.41, p. 59.
  C. Barbu, Teorema lui Smarandache, in his book “Teoreme fundamentale din geometria triunghiului”, Chapter II, Section II.57, p. 337, Editura Unique, Bac?u, 2008.




4)  Generalization of the Bisector Theorem
    --------------------------------------
Let AM be a Cevian of the triangle ABC which forms the angles A1 and A2
with the sides AB and AC respectively.
Then:
        BA     BM   sin A2
       ---- = ----.--------
        CA     CM   sin A1

Reference:
  F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev 
    University Press, Kishinev, Problem 61, pp. 41-42, 1997.
    


5)  Generalization of the Altitude Theorem
    --------------------------------------
Let AD be the altitude of the triangle ABC which forms the angles A1 and A2
with the sides AB and AC respectively.
Then:
        2
      AD  = BD.DC.cot A1.cot A2 

Reference:
  F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev 
    University Press, Kishinev, Problem 62, pp. 42-43, 1997.
    


6)  Collinear Points Theorem
    ------------------------
Let A, B, C, D be collinear points and O a point not on their line.
Then:

        2     2         2     2      
     (OA  - OC )BD + (OD  - OB )AC = 
     
                      2     2     2         3     3     3
     = 2AB.BC.CD + (AB  + BC  + CD )AD - (AB  + BC  + CD )

Reference:
  F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev 
    University Press, Kishinev, Problem 82, p. 61, 1997.


7)  Median Point Theorem
    --------------------
Let P be a point on the median AA' of the triangle ABC.  One notes by
B' and C' the intersections of BP with AC and of CP with AB 
respectively.  Then:
  a) B'C' is parallel to BC.
  b) In the case when AA' is not a median, let A'' be the intersection
of B'C' with BC.  Then A' and A'' divide BC in an anharmonic rapport.   

Reference:
  F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev 
    University Press, Kishinev, Problem 81, p. 60, 1997.