IGO 2016 Medium/2
 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 $CAP$.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
 Thamim Zahin
 Posts: 98
 Joined: Wed Aug 03, 2016 5:42 pm
Re: IGO 2016 Medium/2
Extend the segment $QD$. Name it $C'$.
$\angle CQB=\angle BAC=\angle a$
$\angle BAD=\angle DPB=\angle b$ [cyclic quad]
$\angle DAP=\angle C'DP=\angle a$ [$DC'$ 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, $\angle c+\angle b=\angle a$
Or. $\angle CAD=\angle PAD$
$[Done]$
$\angle CQB=\angle BAC=\angle a$
$\angle BAD=\angle DPB=\angle b$ [cyclic quad]
$\angle DAP=\angle C'DP=\angle a$ [$DC'$ 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, $\angle c+\angle b=\angle a$
Or. $\angle CAD=\angle PAD$
$[Done]$
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I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.