IGO 2016 Medium/2

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

IGO 2016 Medium/2

Unread post by Thamim Zahin » Tue Jan 10, 2017 3:57 pm

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.

User avatar
Thamim Zahin
Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm

Re: IGO 2016 Medium/2

Unread post by Thamim Zahin » Thu Jan 12, 2017 3:45 pm

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. $\angle CAD=\angle PAD$

$[Done]$
Attachments
I2.png
I2.png (48.01 KiB) Viewed 183 times
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.

Post Reply