IMO Shortlist 1983 problem 4
Dodao/la:
arhiva2. travnja 2012. On the sides of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, three similar isosceles triangles
![ABP \ (AP = PB)](/media/m/f/b/0/fb0ba3fd578d1c4248452cbd11ab4855.png)
,
![AQC \ (AQ = QC)](/media/m/5/a/b/5abc97b2c2e8def8228eb09810f575ad.png)
, and
![BRC \ (BR = RC)](/media/m/2/d/a/2dac838f4818ba415e86fd0cf4d0bce1.png)
are constructed. The first two are constructed externally to the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, but the third is placed in the same half-plane determined by the line
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
as the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Prove that
![APRQ](/media/m/d/2/0/d20248f15738d81e5112dfd6597e1b98.png)
is a parallelogram.
%V0
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.
Izvor: Međunarodna matematička olimpijada, shortlist 1983