Let ABC an acute triangle. The circle with diameter the altitude BD meets sides
AB,BC at K,L respectively. The tangents of this circle at K,L meet at M.Prove that
line BM bisects segment AC.
Babis
Nice but not easy !
- karlosson_sul_tetto
- Messaggi: 1454
- Iscritto il: 10 set 2009, 13:21
- Località: Napoli
Well, I did a drawing and I think watching it is always possible, with ABC acute:
Beh, ho fatto un disegno e penso guardandolo che è sempre possibile, con la ABC acutangolo:
Ну,я сделал рисунок и смотря на него думаю что всегда возможно:
Bueno, hice un dibujo y creo que viendo que siempre es posible, con ABC aguda
E pra, unë bëra një vizatim dhe unë mendoj se shikuar se është gjithmonë e mundur, me ABC akut
Nou, ik heb een tekening te kijken en ik denk dat het altijd mogelijk, met ABC acute
Nå, jeg gjorde en tegning, og jeg tror, at se det er altid muligt, med ABC akut
Λοιπόν, έκανα ένα σχέδιο και νομίζω ότι βλέποντας ότι είναι πάντα δυνατό, με οξεία ABC
Nun, ich habe eine Zeichnung und ich denke, gerade ist es immer möglich, mit ABC akuten
Eh bien, j'ai fait un dessin et je pense à regarder, il est toujours possible, avec ABC aiguë
Beh, ho fatto un disegno e penso guardandolo che è sempre possibile, con la ABC acutangolo:
Ну,я сделал рисунок и смотря на него думаю что всегда возможно:
Bueno, hice un dibujo y creo que viendo que siempre es posible, con ABC aguda
E pra, unë bëra një vizatim dhe unë mendoj se shikuar se është gjithmonë e mundur, me ABC akut
Nou, ik heb een tekening te kijken en ik denk dat het altijd mogelijk, met ABC acute
Nå, jeg gjorde en tegning, og jeg tror, at se det er altid muligt, med ABC akut
Λοιπόν, έκανα ένα σχέδιο και νομίζω ότι βλέποντας ότι είναι πάντα δυνατό, με οξεία ABC
Nun, ich habe eine Zeichnung und ich denke, gerade ist es immer möglich, mit ABC akuten
Eh bien, j'ai fait un dessin et je pense à regarder, il est toujours possible, avec ABC aiguë
- Allegati
-
- tr3.jpg (11.38 KiB) Visto 2471 volte
"Inequality happens"
---
"Chissa se la fanno anche da asporto"
---
"Chissa se la fanno anche da asporto"
-
- Messaggi: 1776
- Iscritto il: 17 nov 2007, 19:12
- karlosson_sul_tetto
- Messaggi: 1454
- Iscritto il: 10 set 2009, 13:21
- Località: Napoli
-
- Messaggi: 849
- Iscritto il: 22 ott 2006, 14:36
- Località: Carrara/Pisa
@karlosson: lol
btw let H be the projection of B on CA, so $ \angle LKB = \angle LHB = 90 - \angle LBH = \angle BCA $ and equally we have $ \angle KLB = \angle CAB $, so BLK and BCA are similar. Olso the simmetric of BLK wrt the angle bisector of $ \angle ABC $ is a triangle homotetic with ABC, with an homothety with center B. But BM is the symmedian of BKL as know, so it's also the median of ABC.
I think that this prof is valid also if the triangle is not acute.
btw let H be the projection of B on CA, so $ \angle LKB = \angle LHB = 90 - \angle LBH = \angle BCA $ and equally we have $ \angle KLB = \angle CAB $, so BLK and BCA are similar. Olso the simmetric of BLK wrt the angle bisector of $ \angle ABC $ is a triangle homotetic with ABC, with an homothety with center B. But BM is the symmedian of BKL as know, so it's also the median of ABC.
I think that this prof is valid also if the triangle is not acute.