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### BdMO National Secondary 2006/10

Posted: **Sat Feb 24, 2018 9:42 pm**

by **samiul_samin**

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

### Re: BdMO National Secondary 2006/10

Posted: **Sat Feb 24, 2018 11:49 pm**

by **Tasnood**

Complicated. No $A$ found in the diagram.

### Re: BdMO National Secondary 2006/10

Posted: **Sat Mar 03, 2018 5:11 pm**

by **aritra barua**

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.

### Re: BdMO National Secondary 2006/10

Posted: **Mon Mar 05, 2018 2:03 pm**

by **samiul_samin**

Very nice solution by using

radical axis.Here is the diagram:

Screenshot_2018-03-05-13-58-40-1.png

### Re: BdMO National Secondary 2006/10

Posted: **Tue Mar 06, 2018 3:08 am**

by **mac0220**

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

### Re: BdMO National Secondary 2006/10

Posted: **Mon Feb 18, 2019 11:42 pm**

by **samiul_samin**

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 (13.3 KiB) Viewed 1638 times