The triangle theorem that refuses to be proved the easy way
Take any triangle. If the two lines that bisect two of its angles turn out to be exactly the same length, must the triangle be isosceles, with those two angles themselves equal? It sounds like it should follow from a quick glance, or a line or two of trigonometry.
Reveal the answer
Yes, it's always true, this is the Steiner-Lehmus theorem, first posed by C. L. Lehmus in 1840 and proved soon after by the geometer Jakob Steiner. But every trigonometric proof turns out to require surprisingly heavy machinery, and mathematicians have spent nearly two centuries hunting for a truly simple, elegant synthetic proof using only classical geometry, without ever finding one as short as the question itself.
— C. L. Lehmus (problem, 1840); Jakob Steiner (proof), Steiner-Lehmus theorem — 1840