Which triangles can be split into two isosceles triangles?

(A triangle is isosceles if two of its sides have the same length, or,
equivalently, if two of its angles are equal.)

(This is Macalester College Problem of the Week #867)

There will be a $1 prize for the best correct solution submitted by a
midshipman.

A solution is "correct" if it answers the question correctly and explains
the answer; a solution is "best" if it includes the clearest correct
explanation.  Solutions are due by noon on Thursday, November 12, 1998.
Submit solutions to Prof. Hanna at mathprob@nadn.navy.mil, or via the
mailbox in Chauvenet 301.



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Correct solutions to problem #79, ordering Chicken McNuggets, came from
Midn. 1/c Bessman and Bower, 2/c Chase and Gott, 3/c Christman and Rivera,
and 4/c Patterson, and from non- midshipmen L. B. Baskin and D. Joyner.
Mr. Patterson gets the $1 prize; his solution is posted on the bulletin
board in the Chauvenet lab-deck corridor.  (You cannot buy exactly 43
Chicken McNuggets, but any number larger than that is possible.)