IMO Shortlist 1998 problem G5
Dodao/la:
arhiva2. travnja 2012. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle,
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
its orthocenter,
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
its circumcenter, and
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
its circumradius. Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
be the reflection of the point
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
across the line
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
, let
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
be the reflection of the point
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
across the line
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
, and let
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
be the reflection of the point
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
across the line
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
. Prove that the points
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
are collinear if and only if
![OH=2R](/media/m/d/7/4/d748788aacde8039097f261e14db26a5.png)
.
%V0
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
Izvor: Međunarodna matematička olimpijada, shortlist 1998