## IGO 2016 Medium/2

For discussing Olympiad level Geometry Problems
Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

### IGO 2016 Medium/2

2. Let two circles \$C_1\$ and \$C_2\$ intersect in points \$A\$ and \$B\$. The tangent to \$C_1\$ at \$A\$ intersects \$C_2\$ in \$P\$ and the line \$PB\$ intersects \$C_1\$ for the second time in \$Q\$ (suppose that \$Q\$ is outside \$C_2\$). The tangent to \$C_2\$ from \$Q\$ intersects \$C_1\$ and \$C_2\$ in \$C\$ and \$D\$, respectively (The points \$A\$ and \$D\$ lie on different sides of the line \$PQ\$). Show that \$AD\$ is bisector of the angle \$CAP\$.
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Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

### Re: IGO 2016 Medium/2

Extend the segment \$QD\$. Name it \$C'\$.

\$\angle CQB=\angle BAC=\angle a\$

\$\angle DAP=\angle C'DP=\angle a\$ [\$DC'\$ is tangent to \$(DAP)\$]

Now, notice that \$\angle DAC=\angle b+\angle c\$ and \$\angle DAP=\angle a\$

So, we have to proof that \$\angle c+\angle b=\angle a\$

Now, in triangle \$\triangle DPQ\$, \$DC'\$ is extension of \$QD\$.

So, \$\angle c+\angle b=\angle a\$