ABC is an isosceles triangle in which AB = AC and angle A = 20°.
BD is a line segment intersecting angle B, such that angle DBC = 70°.
A line segment CE intersects angle C, such that angle ECB = 60°.
ED is Joined.

Determine the measure of angle EDB.
Show your work!

Solution to the Problem:

The measure of angle EDB = 20 degrees.

The problem is a classic, named after Edward M Langley, who set it in The Mathematical Gazette, Vol. 11, No. 160 (Oct., 1922), p. 173.  However, the problem is older than this; it appears, for example, in a Cambridge college scholarship examination of 1916.   It appears to be an easy problem, but it is deceivingly difficult.

Since ABC is isosceles, angle ABC = angle ACB.
Since the measures of the 3 angles in a triangle, add up to 180 degrees, angle ABC = angle ACB = 80 degrees and angle ABD = 10 degrees and angle DCE = 20 degrees.

Let x = measure of angle EDB.




Now use the following three triangles and apply the Law of Sines:






Now set the two expressions for BD equal to each other and solve for the value of BE/BC:




Now set the two expressions for BE/BC equal to each other:




Now use the trig identity for the sin(A -B).   Then evaluate the sines and cosines and solve the trig equation:




Here are all the measures of the angles in the triangle:




Correctly solved by:

1. Rob Miles Northbrook, Illinois
2. Kelly Stubblefield Mobile, Alabama
3. Michael Newton Mountain View High School,
Mountain View, Wyoming
4. Brijesh Dave Mumbai city, Maharashtra, India
5. Mikaela Williams Mountain View High School,
Mountain View, Wyoming
6. Tyler Petersen Mountain View High School,
Mountain View, Wyoming


Several other people sent in the correct answer but did not show any work or could not prove it, so those answers were not accepted.

Michael Newton was the first to use AUTOCAD to solve it.

His image from AUTOCAD appears below:



Brijesh Dave sent in the following image: