Triangle A B C; A B = c , B C = a , A C = b; angle bisector of angle ( c, a) cuts A C in point EWhy is the following true? | A E | = b c a + c Where does that come from?

Question

Triangle   A  B  C;   A  B  =  c  ,  B  C  =  a  ,  A  C  =  b; angle bisector of angle (  c,  a) cuts   A  C in point   EWhy is the following true?      |    A  E      |    =            b      c              a      +      c      Where does that come from?

Emely English · Accepted Answer

Hint: We get            A      E              E      C        =      c    a  And  E  C  =  b  −  A  Ethen we get            b      −      A      E              A      E        =      c    a

Katelyn Reyes · Accepted Answer

let   β be half of the bisected angle, and   θ denote the angle AEB.then the sine rule in triangle AEB gives:                    |            A      E              |                    sin      ⁡      β        =      c          sin      ⁡      θ      and the sine rule in triangle CEB gives:                    |            E      C              |                    sin      ⁡      β        =      a          sin      ⁡      (      π      −      θ      )      since   sin  ⁡  θ  =  sin  ⁡  (  π  −  θ  ) and       |    A  E      |    +      |    E  C      |    =  b, the result follows

Triangle ABC; AB=c,BC=a,AC=b; angle bisector of angle (c,a) cuts AC in point E. Why is the following true? |AE|=bc/a+c Where does that come from?

Answered question

Answer & Explanation

New Questions in Elementary geometry