Pythagoras - Right Angle Triangle

Square on the hypotenuse of a right angled triangle is equal to the squares on the other two sides 

ABC is a right angled triangle with ^ABC being the right angle.

Show that the square on AC equals the sum of square on AB and the square on AC.

Consider triangles AGC & ABK

      ^CAG=^CAB + ^BAG

          = ^CAB + rt angle         (HA is a sqare)

          = ^CAB + ^CAK             (AJ is a sqare)

          = ^BAK

       AG = AB                  (Sides of a square)

       AC = AK                  (Sides of a square)

  tri.AGC = tri.ABK            (2 sides & inc. angle)

Draw line BM parallel to AK & CJ cutting KJ at L and AC at N

ABLK is a trapezium as AK is parallel BL

  Tri.ANK = Tri.ABK             (Triangles same base between parallels have = area)

          = Tri.AGC             (Proven above)

          = Tri.GBA             (Tri.s same base between parallels HC & GA have = area)

  Tri.ANK = 1/2 Rect.AL         (Rectangle diag. creates two equal triangles)

  Tri.GBA = 1/2 Rect.GB         (Rectangle diag. creates two equal triangles)

So

  Rect.AL = Square GB

Similarly it can be shown

  Rect.NJ = Square CE

So

                  Square AJ=Square AH + Square EC