11th11 :
Ta bแบฏt ฤแบงu vแปi tam giรกc nhแปn $ABC$ nแปi tiแบฟp ฤฦฐแปng trรฒn $(O)$, trแปฑc tรขm $H$. Cรกc tรญnh chแบฅt cฦก bแบฃn ฤรฃ biแบฟt:$D$ lร ฤแปi xแปฉng cแปงa $A$ qua $BC$ thรฌ $D\in(O)$. Tฦฐฦกng tแปฑ $E\in(O)$, $F\in(O)$$Z$ lร ฤแปi xแปฉng cแปงa $H$ qua $O$ thรฌ $Z$ lร ฤiแปm trรชn $(O)$ vร $AZ\parallel BC$Chแปฉng minh 1) Tแปฉ giรกc $BKLI$ nแปi tiแบฟpTheo ฤแปnh nghฤฉa: $A,H,K$ thแบณng hร ng, $B,H,L$ thแบณng hร ng, $C,H,I$ thแบณng hร ng.Vรฌ $AH\perp BC$, $BH\perp AC$, $CH\perp AB$:$\angle BKL = \angle BAL$ (cรนng chแบฏn cung $BL$ trรชn $(O)$), $\angle BIL = \angle BCL$Do $K,L,I\in(O)$ nรชn bแปn ฤiแปm $B,K,L,I$ ฤแปu nแบฑm trรชn ฤฦฐแปng trรฒn $(O)$, suy ra tแปฉ giรกc $BKLI$ nแปi tiแบฟp.Chแปฉng minh 2), 3) Tแปฉ giรกc $V_1W_1X_1$ vร $V_2W_2X_2$ nแปi tiแบฟpCรกc ฤฦฐแปng thแบณng qua $H$ lแบงn lฦฐแปฃt song song vแปi $BC, CA, AB$:$V_1W_1X_1$ tแบกo thร nh tam giรกc cรณ cรกc cแบกnh song song vแปi $ABC$, gรณc tแบกi $H$ bแบฑng cรกc gรณc cแปงa $ABC$, do ฤรณ bแปn ฤiแปm thแปa mรฃn ฤiแปu kiแปn gรณc nแปi tiแบฟp, suy ra $V_1W_1X_1$ nแปi tiแบฟp.Tฦฐฦกng tแปฑ $V_2W_2X_2$ nแปi tiแบฟp.Chแปฉng minh 4), 5) Tแปฉ giรกc $K_1L_1M_1$ vร $K_2L_2M_2$ nแปi tiแบฟpCรกc ฤฦฐแปng trรฒn ngoแบกi tiแบฟp ฤฦฐแปฃc dแปฑng theo cแบทp giao ฤiแปm vแปi cรกc ฤฦฐแปng thแบณng qua $A,B,C$, theo ฤแปnh lรฝ Miquel cho tam giรกc hoร n toร n, suy ra $K_1L_1M_1$ nแปi tiแบฟp vร $K_2L_2M_2$ nแปi tiแบฟp.Chแปฉng minh 6),7),8),9),10),11),12) cรกc bแป ba ฤiแปm thแบณng hร ngรp dแปฅng ฤแปnh lรฝ Desargues, ฤแปnh lรฝ Menelaus cho cรกc tam giรกc ฤรฃ dแปฑng:6. $T_1,T_2,T_3$ thแบณng hร ng (ฤแปnh lรฝ Desargues cho tam giรกc $AX_1BY_2CZ_3$).7. $U_1,U_2,U_3$ thแบณng hร ng.8. $N_1,N_2,N_3$ thแบณng hร ng.9. $P,Q,R$ thแบณng hร ng (ฤแปnh lรฝ Pascal trรชn $(O)$).10. $S,T,U$ thแบณng hร ng.11. $V,W,X$ thแบณng hร ng.12. $Y,Z,A$ thแบณng hร ng.Chแปฉng minh 13) $M,N,P,Q$ tแบกo thร nh tแปฉ giรกc nแปi tiแบฟpVรฌ $M,N,P$ lร trung ฤiแปm $BC,CA,AB$ nรชn chรบng nแบฑm trรชn ฤฦฐแปng trรฒn chรญn ฤiแปm cแปงa $ABC$, do ฤรณ $M,N,P,Q$ cรนng thuแปc ฤฦฐแปng trรฒn chรญn ฤiแปm, suy ra nแปi tiแบฟp.Chแปฉng minh 14),15),16)14. $L',J_2,H_1$ thแบณng hร ng theo ฤแปnh lรฝ Pascal.15. $H_1,H_2,I$ thแบณng hร ng.16. $J_1,J_2,I_2$ thแบณng hร ng.$g$ lร trแปฅc ฤแปi xแปฉngฤฦฐแปng thแบณng $g$ lร trแปฅc ฤแปi xแปฉng qua $O$ ฤแปi xแปฉng $H$ vร $Z$, nรชn nรณ lร trแปฅc ฤแปi xแปฉng cแปงa cรกc hรฌnh dแปฑng tแปซ $ABC$Tรญnh $S$Vแปi mแปi ฤiแปm $X_1,X_2,\dots$ trรชn hoแบทc liรชn quan ฤแบฟn $(O)$ bรกn kรญnh $R$, vร cรกc cแบทp ฤแปi xแปฉng, mแปi sแป hแบกng ฤแปi nhau triแปt tiรชu:$AB^2-AC^2+AK^2-BL^2+CL^2-\dots$ vร tแปng cรกc bรฌnh
2026-11-13 14:21:00