BdMO National Secondary 2006/10

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
User avatar
samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

BdMO National Secondary 2006/10

Unread post by samiul_samin » Sat Feb 24, 2018 9:42 pm

Two circles with the centers $O$ and $C$,touch each other externally at $F$.The tangent $TT'$ touches two circles at $P$ and $Q$ respectively.Prove $\angle PFO+\angle QFC=1$ right angle
Screenshot_2018-02-23-20-04-06-1.png
Last edited by samiul_samin on Sun Feb 25, 2018 7:00 am, edited 1 time in total.

User avatar
Tasnood
Posts: 73
Joined: Tue Jan 06, 2015 1:46 pm

Re: BdMO National Secondary 2006/10

Unread post by Tasnood » Sat Feb 24, 2018 11:49 pm

Complicated. No $A$ found in the diagram.

aritra barua
Posts: 57
Joined: Sun Dec 11, 2016 2:01 pm

Re: BdMO National Secondary 2006/10

Unread post by aritra barua » Sat Mar 03, 2018 5:11 pm

It suffices to prove that $\bigtriangleup PFQ$ is a right triangle.Now,let $OF$ and $CF$ extended meet the $2$ circles for the second time at $A$ and $B$ respectively.$\bigtriangleup APF$ and $\bigtriangleup BQF$ both are right triangles. So,by Alternate Segment Theorem,angle $FPQ$=angle $FAP$=$90-AFP$.Let the radical axis of the $2$ circles meet $PQ$ at $M$.It is well known that $M$ is the midpoint of $PQ$.Since angle $AFM$=$90$°,angle $PFM$=$90-AFP$=angle $FPM$.We therefore have $PM=QM=FM$,so $M$ is the circumcenter of $\bigtriangleup PFQ$,hence $\bigtriangleup PFQ$ is a right triangle.The rest is trivial.

User avatar
samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

Re: BdMO National Secondary 2006/10

Unread post by samiul_samin » Mon Mar 05, 2018 2:03 pm

Very nice solution by using radical axis.Here is the diagram:
Screenshot_2018-03-05-13-58-40-1.png

mac0220
Posts: 4
Joined: Mon Feb 05, 2018 3:45 am

Re: BdMO National Secondary 2006/10

Unread post by mac0220 » Tue Mar 06, 2018 3:08 am

If I just connect O and P , C and Q , they're both perpendiculer on TT'
Now solving the problem will be a whole lot easier :D

User avatar
samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

Re: BdMO National Secondary 2006/10

Unread post by samiul_samin » Mon Feb 18, 2019 11:42 pm

samiul_samin wrote:
Sat Feb 24, 2018 9:42 pm
Two circles with the centers $O$ and $C$,touch each other externally at $F$.The tangent $TT'$ touches two circles at $P$ and $Q$ respectively.Prove $\angle PFO+\angle QFC=1$ right angle Screenshot_2018-02-23-20-04-06-1.png
Screenshot_2019-02-18-23-33-59-1.png
Screenshot_2019-02-18-23-33-59-1.png (13.3 KiB) Viewed 1644 times

Post Reply