# Find the perimeter of $\triangle AMN$

In the triangle ABC, $MN \parallel BC$, $BO$ and $CO$ are angle bisectors of $\angle MBC$ and $\angle NCB$ respectively. If $AB=12$, $BC=24$ and $AC=18$, then what is the perimeter of $\triangle AMN$ ? I tried with similarity but got nothing there. Maybe i'm not seeing the similar triangles in the correct way. Any hints in this kind of problem?

Thanks.

• Hint: triangles $\triangle AMN$ and $\triangle ABC$ are similar with ratio = (height - inradius) / height from $\,A\,$. – dxiv Jun 23 '18 at 3:09
• Hint: O is the incenter, and triangle AMN is similar to triangle ABC – user547075 Jun 23 '18 at 3:13

It's better the following way: $$P_{\Delta AMN}=AM+MO+AN+NO=AM+MB+AN+NC=12+18=30.$$

• Thanks,i found another way to do it and got 30 too!. But thanks anyway. – Rodrigo Pizarro Jun 23 '18 at 3:30
• You are welcome! – Michael Rozenberg Jun 23 '18 at 3:31
• @Rodrigo Pizarro I added something better. – Michael Rozenberg Jun 23 '18 at 3:37
• Exactly the way i did it. Nice. – Rodrigo Pizarro Jun 23 '18 at 3:43

The semi-perimeter is $s = {12+18+24 \over 2} = 27$

The area of $\triangle$ ABC is, by Heron, $\sqrt{27 . 15 . 9 . 3 } = 27 \sqrt{15}$

The inradius is $r = {Area \over s} = \sqrt{15}$ = the elevation of O.

The elevation of A is ${2Area\over 24 } = {54\over 24 }\sqrt{15}$

The similarity ratio comes out from the ratio of the two elevations $= {54\over (54-24) }$

The semi-perimeter of $\triangle$ AMN is $= {30 \over 54} 27 = 15$

The required perimeter is then $30$.