Thread: 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.

Have you drawn a picture to help solve this problem? After doing this label the angles with the information given. Use the fact that a triangle has 180 degrees and the sum of angles around P is 360. Label your any unknowns (there should be 2) and solve them by setting up some simultaneous equations

i dont think i can use simultaneous equations to solve this.

Hello, ahm0605!

. .



Very difficult problem, don't know where to start.
Of course you do . . . Make a sketch!

I don't think the proof is possible.

Code:

                A
                o
                *
               ***
                *
              * * *
                *
             *  *  *
             10 * 40
            *   *   *
                *
           *    *    *
                *
          *     *     *
                *
         *      o      *
            150 * 110
        *     *   *     *
         20 *  100  * 30
       *  *           *  *
        *  x         y  *
    B o   *   *   *   *   o C

And all we have is: .

I have drawn a sketch but don't know where to go from there. I think it's something to do with the cosine rule

This has been bothering me! If we don't crack it soon, I might seek "outside assistance"...

Is anyone listening?

With the given information, ANY triangle is possible.
(Well, with the exception of an equilateral.)

Here is a possible right triangle.

Code:

      A
      ♥
      *◊*
      * ◊ *
      *  ◊  *
      *10 ◊ 40*
      *    ◊    *
      *     ♥     *
      *    ◊ p      *
      *20 ◊       ◊ 30*
      *  ◊              *
      * ◊               ◊ *
      *◊ 70           10    *
    B ♥  *  *  *  *  *  *  *  ♥ C

Soroban has already told you all you need to know- you can't prove it- it is NOT true!

I just wanted to see him draw another picture...

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