Given: triangle ABC with D on AC and E on BC,
angle CAE = 10°,
angle BAE = 70°,
angle CBD = 20°, and
angle ABD = 60°.

Find angle AED using only elementary geometry (no sines, cosines , or any other trigonometry.)

This is a fantastic problem!!!

In the following, three capital letters denote

an angle.

   I used the sine rule multiple times to derive

a formula for AED in terms of arbitrary values

for ABD, BAE, CAE, and CBD.

   It gave the correct values for AED in

Geometer's Sketchpad most of the times

(I'm still working on the other times).

   It gave AED = 20.00000 when I specified

the four values of our problem. 

   For our problem without using trig. I could

find AED = 20 (with proof) only if I could

prove the following lemma:

           The foot of the perpendicular from

        point P to side BC is the midpoint of

        line segment D'E.

           Point P is the intersection of line

        segments AE and D'E'. Points D' and E'

        are the reflections of points D and E

        about the bisector of ACB.

   I'm still trying to prove the lemma. It seems

to hold in Geometer's Sketchpad. Any hints

or help would be appreciated.

   One point I would like to make: I would not even

try this problem without  Geometer's Sketchpad.

Posted by Bractals on 2016-11-16 15:31:08