IMO Shortlist 1977 problem 8
Dodao/la:
arhiva2. travnja 2012. Let
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be a convex quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
and
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
a point inside it. The feet of the perpendiculars from
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
to
![AB, BC, CD, DA](/media/m/3/a/b/3abd768bdbbdf703d6007c48228acd32.png)
are
![A_1, B_1, C_1, D_1](/media/m/f/5/2/f52e1f7cc1c96747cb0d449d3a0347b6.png)
respectively. The feet of the perpendiculars from
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
to the sides of
![S_i](/media/m/9/f/d/9fd19263a5663142484a86f39fb49833.png)
, the quadrilateral
![A_iB_iC_iD_i](/media/m/7/d/d/7dd256b38a88f148ed479331c44c5852.png)
, are
![A_{i+1}B_{i+1}C_{i+1}D_{i+1}](/media/m/e/b/8/eb82bd33716b197ca68b5fbb225e91ff.png)
, where
![i = 1, 2, 3.](/media/m/0/e/0/0e052a0c7d40fa310b3674f924ea5b94.png)
Prove that
![S_4](/media/m/0/e/7/0e70eeca2a9cf1d6d5277d9b296c8f11.png)
is similar to S.
%V0
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.
Izvor: Međunarodna matematička olimpijada, shortlist 1977