Thale's Theorem

Homework for Monday January 30, 2012

1)  ABC is an acute and scalene triangle.
     The circle of diameter [BC] cuts the sides [AB] and [AC] at E and F respectively.
     The circle of diameter [EF] cuts the sides [AB] and [AC] at M and N respectively.

      a)  Draw a figure.
      b)  Show that (BF) is parallel to (EN).
      c)  Show that the ratio AM / AF equals the ratio AE / AB and AM / AE equals AF / AC.
      d)  Deduce that AM x AC = AN x AB
      e)  Show that (MN) is parallel to (BC).


2)  ABCD is a parallelogram such that AB=8cm and AD=4.5cm.
      E is a point on [DA) such that DE=6cm. [EC) cuts [AB] at point I.

      a) Calculate AI,
      b) Construct point J on [DC) such that  DJ=three- fourth of DC.
      c) Show that (AJ) is parallel to (EC).

3)   BIEN is a parallelogram such that BI= 6cm and IE= 9cm and BE= 12cm.
      Let V be a point [BI] such that BV= 4cm.
     The parallel to [IE] passing through V cuts (BE) at D.

     a)  Show that BD= 8cm.
     b)  R is a point of [BN] such that BR= 6cm.
          Show that (DN) is parallel to (EN).
    c)  The parallel to (BE) passing through R cuts (IE)    at K and (NE) at H.
         Calculate HE and EK.

4)  ABC is a scalene triangle.
     a)  M is a point of [AC] such that the ratio CM o verCA =one- third. Construct point M.
     b)  Place point  D symmetric of B with respect to C.
          What does point M represent for triangle AB
     c) The parallel through M to (BD) cuts [AB] and [AD] at F and E respectively.
         Find   4 ratios equal to AM over AC.
         Deduce that M is the midpoint of [EF].