Question 568819
I need to solve a proof for triangle ABC and DEC trying to prove C is the midpoint of AD. I am given that BC is equal to EC and angle A is equal to angle D
<pre>
I'm sorry, but that is not true.  Here is why:

Draw an isosceles triangle, label the two base angles A and D

{{{drawing(200,200,-11.5,11.5,-2,21, locate(-10,0,A), locate(10,0,D),
line(-10,0,10,0),line(-10,0,0,20), line(0,20,10,0))}}}

Pick a point on the base of that isosceles triangle that is
obviously NOT its midpoint, and label it C.

{{{drawing(200,200,-11.5,11.5,-2,21, locate(-10,0,A), locate(10,0,D),
line(-10,0,10,0),line(-10,0,0,20), line(0,20,10,0),
locate(-3,0,C),circle(-3,0,.4)

)}}}

With a compass, swing an arc with center C that cuts both legs of the
isosceles triangle.  Label those two points where the arc cuts those legs 
B and E 

{{{drawing(200,200,-11.5,11.5,-2,21, locate(-10,0,A), locate(10,0,D),
line(-10,0,10,0),line(-10,0,0,20), line(0,20,10,0),
locate(-3,0,C),circle(-3,0,.4), red(arc(-3,0,2*12.80624847,-2*12.80624847,30,120)), locate(5,12,E),locate(-4.5,14.5,B)


)}}}

Draw radii BC and EC

{{{drawing(200,200,-11.5,11.5,-2,21, locate(-10,0,A), locate(10,0,D),
line(-10,0,10,0),line(-10,0,0,20), line(0,20,10,0),
locate(-3,0,C),circle(-3,0,.4), red(arc(-3,0,2*12.80624847,-2*12.80624847,30,120)), locate(5,12,E),locate(-4.5,14.5,B),
line(-10,0,-3.604001604,12.79199679),

line(5,10,-3,0),

line(-3,0,-3.604001604,12.79199679) )}}}


Now in that figure we have triangles ABC and DEC

BC is equal to EC because 
they are radii of the same arc.

Also angle A is equal to angle D because they are base 
angles of an isosceles triangle.

<u>But C is NOT the midpoint of AD!!!</u>

So what you were asked to prove is just not true!!!

Edwin</pre>