116. The length of AE (nine methods !).
by Virgil Nicula, Sep 12, 2010, 9:47 PM
Quote:
The 
-bisector 
meets again the circumcircle 
of the triangle 
at 
. Ascertain the length of the segment ![$[AE]$](//latex.artofproblemsolving.com/0/e/e/0ee81ab9421e97824f7785dc2c44c0f5b04a03b1.png)
.
-bisector 
meets again the circumcircle 
of the triangle 
at 
. Ascertain the length of the segment ![$[AE]$](http://latex.artofproblemsolving.com/0/e/e/0ee81ab9421e97824f7785dc2c44c0f5b04a03b1.png)
.Method 1. Observe that
. Apply the Ptolemeus' relation to the cyclicalquadrilateral
because 
. In conclusion, 
.Method 2. Suppose w.l.o.g. that
. Denote 
. Observe that 
, 
and
because they are right and 
, 
. Thus, 
, 
and
. In conclusion, .Method 3. Denote the diameter
![$ [AS]$](http://latex.artofproblemsolving.com/f/8/4/f842ede4c52003071311b54e4b58c3e670f09f32.png)
of the circle 
and 
, 
. Observe that 
with common value
, i.e. the triangle 
is 
- isosceles. Since 
obtain 
, i.e.
. Since 
, 
obtain 
, i.e. .Method 4. IF
, then 
. Thus, 
,i.e.
. But 
. In conclusion, 
.Method 5.
.Thus,
. In conclusion, .Method 6.
, i.e. .Method 7. Denote
for which 
. Thus, 
and 
, i.e. 
.Therefore,
. In conclusion, .Method 8. Denote
for which 
. Observe that 
and 
.Since
obtain 
, i.e. .Method 9. Denote
for which 
. Observe that 
what means the triangle 
is 
- isosceles. Thus,for the midpoint
of ![$ [CK]$](http://latex.artofproblemsolving.com/6/f/b/6fb3d7b9de1a146d3d5f72c713450188c10795f5.png)
we have 
and 
and 
.Remark. You can obtain directly the relation
from the 
- isosceles triangle 
!See length of angle-bisector (<== click).
This post has been edited 7 times. Last edited by Virgil Nicula, Nov 23, 2015, 7:47 AM
August 2018
January 2018