Thread: Geometry puzzle

Code:

         A 
 
 
           E 
      D 
 
 
 
    C          B

Triangle ABC is isosceles, equal angles at B and C = 80.

D is on AC, such that CD = BC.

E is on AB, such that angle BCE = 60.

Find size of angle DEC. No trigonometry allowed! Have fun.

Originally Posted by Wilmer

Code:

         A 
 
 
           E 
      D 
 
 
 
    C          B

Triangle ABC is isosceles, equal angles at B and C = 80.

D is on AC, such that CD = BC.

E is on AB, such that angle BCE = 60.

Find size of angle DEC. No trigonometry allowed! Have fun.

DEC=30

Full solution:

Spoiler:

I drew a scale diagram and measured it.

BUT your "pencil" you "drew" with could be wide enough to be off a degree or so !!
Nice try, Alex.
But now that you have a diagram to play with, go "get the proof"

1st hint: place point F on AB such that CF = CB

2nd hint: examine triangle CFD

Let be the reflection of across the foot of perpendicular of to . Then , . We find that so . At the same time , and so is equilateral . We thus have . Therefore , is the circumcentre of so

Nice. I like this solution (# means angle) :

Triangle BCE: it is given that #ABC = 80 and #BCE = 60; then #BEC = 40

Triangle CDF: this triangle being equilateral, then #CFD = 60;
since #BFC = 80, this leaves #DFE = 180-80-60 = 40

Triangle DEF: #EDF = #DEF, so #CED + 40 = 70, hence #CED = 30