Točno
19. prosinca 2022. 13:05 (1 godina, 6 mjeseci)
Let
be an isosceles triangle with
, whose incentre is
. Let
be a point on the circumcircle of the triangle
lying inside the triangle
. The lines through
parallel to
and
meet
at
and
, respectively. The line through
parallel to
meets
and
at
and
, respectively. Prove that the lines
and
intersect on the circumcircle of the triangle
.
comment
(According to my team leader, last year some of the countries wanted a geometry question that was even easier than this...that explains IMO 2003/4...)
[Note by Darij: This was also Problem 6 of the German pre-TST 2004, written in December 03.]
Edited by Orl.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![AC=BC](/media/m/b/f/9/bf9c620017f7422049d047d9cfa2af8e.png)
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![AIB](/media/m/b/9/c/b9cd006b91d5057a9510cec1d9c03b15.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
![CB](/media/m/d/d/7/dd7ebd02df3c940a7e47b8a09480e1b1.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
![CB](/media/m/d/d/7/dd7ebd02df3c940a7e47b8a09480e1b1.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![DF](/media/m/3/d/d/3dd8b7899102ac0f0d215a5d87897f88.png)
![EG](/media/m/a/b/5/ab55e009d6dc385fd617dc2306f27e89.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
comment
(According to my team leader, last year some of the countries wanted a geometry question that was even easier than this...that explains IMO 2003/4...)
[Note by Darij: This was also Problem 6 of the German pre-TST 2004, written in December 03.]
Edited by Orl.
Upozorenje: Ovaj zadatak još niste riješili!
Kliknite ovdje kako biste prikazali rješenje.
Kliknite ovdje kako biste prikazali rješenje.
Neka je presjek
i
. Dokazat ćemo da je
na
i na
. (Motivacija za to je činjenica da su
i
homotetični pa se
i
sijeku u jednoj točki, pa ako zadatak vrijedi to će morati biti traženi presjek.)
Zbog simetrije dovoljno je da je na
. Imamo:
, odakle je
tetivan, a kako je
jednakokračni trapez i
leži na toj kružnici. Sličan chase daje i tetivnost
. Sada,
što smo i trebali.
P.S. Dosta lagano za jedan G5, nakon nacrtane slike mislim da se i bez ovog argumenta homotetijom može naslutiti gdje će biti ta točka presjeka i od tamo je chase dosta lagan. Ako se usmjere kutevi može se dobiti da zadatak vrijedi i za izvan