IMO Shortlist 1988 problem 23
Dodao/la:
arhiva2. travnja 2012. Let
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
be the centre of the inscribed circle of a triangle
![ABC.](/media/m/c/b/7/cb77700b4adade65e440645391a8d2ad.png)
Prove that for any point
where
![a = BC, b = CA](/media/m/d/1/a/d1ab8522f03f0085af838a7bb81febc9.png)
and
%V0
Let $Q$ be the centre of the inscribed circle of a triangle $ABC.$ Prove that for any point $P,$
$$a(PA)^2 + b(PB)^2 + c(PC)^2 = a(QA)^2 + b(QB)^2 + c(QC)^2 + (a + b + c)(QP)^2,$$
where $a = BC, b = CA$ and $c = AB.$
Izvor: Međunarodna matematička olimpijada, shortlist 1988