20-30-130 Triangle Problem

Given triangle ABC with an angle of  20  degrees at  A  bisected by   AF  and an angle of 30 degrees at B cut by a segment   BE   such that angle ABE is  10 degrees and angle CBE is 20 degrees.   What is the measure of angle DCE?

GSP file



Hints: Warning, these may not be helpful!!! GSP files are provided.


Hint 1


Hint 2

Now can we show that the added lines to this figure embed the original problem?


Hint 3

Since we know that AD = DB, take D as the center of a circle and extend lines from the triangle to explore sizes of subtended angles:

It is very tempting to see the measure of Arc IJ to be 20 degrees and hence the measure of arc IL to be 120 degees. That would mean that angle CDE is 120 degrees and therefore in triangle CDE we would know two of the angles, 120 degrees and 30 degrees, so the other angle, Angle DCE, is 30 degrees. HOWEVER, can we prove arc IJ has a measure of 20 degrees?