Similarity and right angle triangle

 Theorem : in a right angle triangle, if the altitude is drawn to the hypotenuse,then the two Triangles formed are similar to the original triangle and to each other.

Given :  in △ ABC,   ∠ABC=90º, 

        SegBD ⊥ SegAC,        A-D-C

TO prove:          △ABD   ~   △ABC

                              △  BDC  ~    △ ABC

                              △  ADB    ~   △  BDC

Proof: in   △  ADB and    △  ABC

        △ DAB  ≅  △ BAC........ (common angle)

        △ ADB  ≅   △ ABC......... (each90º)

        △ ADB  ~  △ ABC............... (AA Test)..... (1)

Now in BDC and ABC

       △ BCD  ≅  △ ACB............. (common angle)

      △  BDC   ≅  △  ABC......... ( each 90º)

      △ BDC  ~  △ ABC................ (AA Test )...... (2)

from equation (1) and (2)

      △ ADB  ~   △ BDC.............                          (3)

from equation (1),(2) and(3)

   △ ADB  ~  △ BDC  ~ △ABC..... (Transitivity)