IMO Shortlist 1982 problem 17
Dodao/la:
arhiva2. travnja 2012. The right triangles
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and
![AB_1C_1](/media/m/c/9/1/c913e170c9a0252fb1f116be26381513.png)
are similar and have opposite orientation. The right angles are at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
, and we also have
![\angle CAB = \angle C_1AB_1](/media/m/d/4/6/d46fcb4a0cedeb0dbff0e6b047843041.png)
. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the point of intersection of the lines
![BC_1](/media/m/7/f/a/7fa32034355e0e5570f9d617a0b8ddd7.png)
and
![B_1C](/media/m/d/f/0/df028f1c7d0cd45d1fd5e8abd627016a.png)
. Prove that if the lines
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
and
![CC_1](/media/m/0/8/0/0801dadadab90cd497baf071c549e706.png)
exist, they are perpendicular.
%V0
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $\angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.
Izvor: Međunarodna matematička olimpijada, shortlist 1982