Verrrry interesting !
Look at the drawing:
JX = JY, so the inside triangle is isosceles.
Angle 1 = Angle 2.
Base angles of an isosceles triangle are equal.
Angles (1 + 3) = Angles (2 + 4).
The sum of all the angles on one side of a line is 180 degrees.
Well ! If (1 + 3) = (2 + 4), and 1 = 2, then Angle 3 = Angle 4 .
Now look at the two skinny, slanty outside triangles:
JX = JY given
Angle 3 = Angle 4
KX = KY given
So the two skinny slanty outside triangles are congruent (side-angle-side)
Then JK = JL corresponding parts of congruent triangles.
So the big outside triangle is isosceles, since JK = JL .
Q.E.D.