Točno
23. srpnja 2017. 20:58 (6 godine, 11 mjeseci)
In a plane the circles
and
with centers
and
, respectively, intersect in two points
and
. Assume that
is obtuse. The tangent to
in
intersects
again in
and the tangent to
in
intersects
again in
. Let
be the circumcircle of the triangle
. Let
be the midpoint of that arc
of
that contains
. The lines
and
intersect
again in
and
, respectively. Prove that the line
is perpendicular to
.
![\mathcal K_1](/media/m/0/5/a/05a00b250f840d61d895f7d9b92f7210.png)
![\mathcal K_2](/media/m/5/9/9/599bc1a1fd41389ed94ca196d29d3eef.png)
![I_1](/media/m/4/1/6/416fe38d243ecd6d66747e4d88b6518d.png)
![I_2](/media/m/c/b/0/cb0739259063636b82f7b6b3e9dd3de0.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![\angle I_1AI_2](/media/m/9/c/0/9c0ede57b3e373a6d8ac7da550b2c57c.png)
![\mathcal K_1](/media/m/0/5/a/05a00b250f840d61d895f7d9b92f7210.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![\mathcal K_2](/media/m/5/9/9/599bc1a1fd41389ed94ca196d29d3eef.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![\mathcal K_2](/media/m/5/9/9/599bc1a1fd41389ed94ca196d29d3eef.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![\mathcal K_1](/media/m/0/5/a/05a00b250f840d61d895f7d9b92f7210.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![\mathcal K_3](/media/m/5/9/5/595d793b62d10dc6877f4191ec240c3d.png)
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![\mathcal K_3](/media/m/5/9/5/595d793b62d10dc6877f4191ec240c3d.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![\mathcal K_3](/media/m/5/9/5/595d793b62d10dc6877f4191ec240c3d.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
![AE](/media/m/c/e/3/ce31f42a92358c211bccb23e6a92fb55.png)
![KL](/media/m/b/1/a/b1ab64b407588444cd365224f9b482b4.png)
Upozorenje: Ovaj zadatak još niste riješili!
Kliknite ovdje kako biste prikazali rješenje.
Kliknite ovdje kako biste prikazali rješenje.
![\text{Lema o spiralnoj sličnosti}](/media/m/5/5/2/552dae00e22f258d740240251cee44a6.png)
![A,B,C,D](/media/m/8/5/d/85d135de173dbb765c7a2f175c5b2f60.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
![(ABX)](/media/m/7/2/2/722692b01d6e25e897f4fc8d8ba0605c.png)
![(CDX)](/media/m/d/7/d/d7d6f30ea541724f53271edd4f0c2da2.png)
![h](/media/m/e/4/3/e438ac862510e579cf5cbdbe5904d4ba.png)
![h(A)=C, h(B)=D](/media/m/a/9/a/a9a7571ce663aacce339e93e057bad0f.png)
![\text{Yébalatheova lema}](/media/m/d/3/d/d3d5ee182a1306cddce0a145fd2bd0fc.png)
![\omega_1, \omega_2](/media/m/3/8/4/3844c8df3d9bcd2127f1839c4b9ef28c.png)
![O_1, O_2](/media/m/f/8/6/f86e91464bd9fee950bb9c887390edf6.png)
![I_1, I_2](/media/m/4/c/f/4cfac35f98eec866959b84a06730617b.png)
![O_1 O_2 \perp I_1 I_2](/media/m/5/d/f/5df72e2283288c129bfee3317e63073a.png)
Neka je
![\angle DAC = \alpha, \angle ADC = \delta, \angle DCA = \gamma, \angle DBA = \chi](/media/m/e/0/9/e0997e1040c137c0a73f120c3af92abf.png)
Primjenom (*) za
![A \equiv X \equiv C](/media/m/d/c/a/dca13ccd9b72119cd48b6581f1b7618a.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![\angle DBA = \angle BCA](/media/m/c/b/3/cb34537e56cbe2f39e0fb15c74864e76.png)
![\angle BAD= \angle ACB, \angle BDA= \angle BAC](/media/m/2/4/f/24f5647a76cd694fa9c49e397b104601.png)
![DBCA](/media/m/7/b/9/7b90a27d2117623bb0bcb333d420763e.png)
![\chi = 180 - \alpha](/media/m/f/8/6/f86f23180c9fa31c63c43fe6553c3dc7.png)
![\angle DBC = 2\alpha = \angle DEC](/media/m/f/5/4/f54e2cad79c865a7142d70231850dd60.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![(ACD)](/media/m/3/a/e/3aed0b79a806b8743d38dc39c4599a2a.png)
Nadalje neka je
![I_3](/media/m/9/d/8/9d8b7d345598e5f924b2b2d7b253b2aa.png)
![\mathcal K_3](/media/m/5/9/5/595d793b62d10dc6877f4191ec240c3d.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![KL](/media/m/b/1/a/b1ab64b407588444cd365224f9b482b4.png)
![EI_3 \perp CD](/media/m/5/f/9/5f948c02596c6b6e0a4f35cb36ebc48c.png)
![AE \perp KL](/media/m/5/5/1/551ab61a197f9e07d95757247dfb2c4e.png)
![\angle(AE,EI_3)=\angle(DC,KL)](/media/m/5/a/1/5a1c4da3d35ebf296abe591054ed57fd.png)
![\angle AEI_3 = 180-\angle DMK](/media/m/2/6/a/26a6bf76bff0711a50c64a2c9fcaabf4.png)
To lako dovršavamo angle chasingom. Kako je
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![(ACD)](/media/m/3/a/e/3aed0b79a806b8743d38dc39c4599a2a.png)
![\angle AEI_3 = 2\gamma + \alpha](/media/m/8/1/a/81abb09fe3b23fb6506945c5e89e57fa.png)
![DMK](/media/m/f/3/a/f3a91ff93fde20745ca658f82d791006.png)
![\angle DMK=\delta-\gamma=180-\alpha-\gamma-\gamma=180-\alpha-2\gamma=180-\angle AEI_3](/media/m/3/0/7/307ee321edb137dd1ab9b83cd4b4abd7.png)
![\square](/media/m/d/1/5/d15a79517835696a16bf5fe7bd240ef9.png)
Ocjene: (1)
Komentari:
tuksy, 8. kolovoza 2017. 19:19
Poštovani kolega rhldj,
vaše rješenje uistinu jest točno, međutim izgleda da ste pri rješavanju zamijenili K i L. Malen previd, ukoliko bi uz rješenje bila priložena skica bio bi potpuno zanemariv, međutim u ovakvom formatu to postaje znatan problem. Također ste napravili jedan manji tipfeler; naime napisali ste
DBA =
BCA umjesto
DBA =
ABC. Također, čini se da ste bez objašnjenja pretpostavili
-
> 0? Opet, nije velika greška jer samo nedostaje kratko objašnjenje zašto to možemo pretpostaviti. Također, u sljedećoj rečenici loše je definirano kako se primjenjuje lema (*): "Primjenom (*) za A
X
C znamo da je B centar spiralne sličnosti koja šalje D u A i A u C."
Rješenje je točno, ali molim da ispravite navedene greške i poradite na svom zapisivanju rješenja.
S poštovanjem,
Dipl. ing. MEMO-a tuksy
vaše rješenje uistinu jest točno, međutim izgleda da ste pri rješavanju zamijenili K i L. Malen previd, ukoliko bi uz rješenje bila priložena skica bio bi potpuno zanemariv, međutim u ovakvom formatu to postaje znatan problem. Također ste napravili jedan manji tipfeler; naime napisali ste
![\angle](/media/m/d/6/5/d65ab0c754453c498c44449ead23bd1a.png)
![\angle](/media/m/d/6/5/d65ab0c754453c498c44449ead23bd1a.png)
![\angle](/media/m/d/6/5/d65ab0c754453c498c44449ead23bd1a.png)
![\angle](/media/m/d/6/5/d65ab0c754453c498c44449ead23bd1a.png)
![\delta](/media/m/6/4/2/642590ff3ac7e36d1b8de757ce09c701.png)
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
![\equiv](/media/m/d/1/1/d11a1813acec7c47ba16da45162ba1a6.png)
![\equiv](/media/m/d/1/1/d11a1813acec7c47ba16da45162ba1a6.png)
Rješenje je točno, ali molim da ispravite navedene greške i poradite na svom zapisivanju rješenja.
S poštovanjem,
Dipl. ing. MEMO-a tuksy