Prove that triangle ABC is isosceles

Triangle ABC has the following properties: there is an interior point P such that angle PAB = 10 degrees, angle PBA = 20 degrees, angle PCA = 30 degrees and angle PAC = 40 degrees. Prove that triangle ABC is isosceles.

Very difficult problem don't know where to start.

Any help appreciated.

Re: Prove that triangle ABC is isosceles

Sorry to open the old thread, but a user has just posted this problem recently, and I think i should give a link to the solution, as the problem is both famous, interesting and not wrong...

mathhelpforum.com/new-users/212009-mongolias-province-math-competition-9th-grade.html#post765883

the link does not seem to work:

consider and we know , hence

now we have to use the trigonometric version of ceva's theorem:

=>

=>

=> [#using product-sum formula]

=>

=>

so as , we can write or

hence ,therefore triangle ABC is isosceles