IGO 2016 Medium/4

For discussing Olympiad level Geometry Problems
User avatar
Thamim Zahin
Posts:98
Joined:Wed Aug 03, 2016 5:42 pm
IGO 2016 Medium/4

Unread post by Thamim Zahin » Tue Jan 10, 2017 4:03 pm

4. Let $w$ be the circumcircle of right-angled triangle $ABC (\angle A = 90)$. Tangent to $w$ at point $A$ intersects the line $BC$ in point $P$. Suppose that $M$ is the midpoint of (the smaller) arc $AB$, and $PM$ intersects $w$ for the second time in $Q$. Tangent to $w$ at point $Q$ intersects $AC$ in $K$. Prove that $\angle PKC = 90$.
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/4

Unread post by Thamim Zahin » Mon Jun 05, 2017 10:23 pm

Suppose $AB<AC$ (the solution is same if $AB>AC$)
Here, $\triangle KAQ \sim \triangle KCQ \Rightarrow \frac{KA}{KC}=\frac{[KAQ]}{[KQC]}=(\frac{AQ}{QC})^2$

And, $\triangle PAB \sim \triangle PAC \Rightarrow \frac{PB}{PC}=\frac{[APB]}{[APC]}=(\frac{AB}{AC})^2$

We also have, $\triangle PQA \sim \triangle PAM \Rightarrow \frac{AQ}{AM}=\frac{PA}{PM} ...(i)$

And, $\triangle PBM \sim \triangle PQC \Rightarrow \frac{BM}{QC}=\frac{PM}{PC} ...(ii)$

By multiplying $(i)$ and $(ii)$ $\Rightarrow \frac{AQ}{AM}\times\frac{BM}{QC}=\frac{PA}{PM}\times\frac{PM}{PC}\Rightarrow\frac{AQ}{QC}=\frac{PA}{PC}$ [because $MA=MB$ ]

It is known that $\triangle PAB \sim \triangle PCA \Rightarrow \frac{PA}{PC}=\frac{AB}{AC}$
So, $\frac{AQ}{QC}=\frac{AB}{AC} \Rightarrow (\frac{AQ}{QC})^2=(\frac{AB}{AC})^2 \Rightarrow \frac{KA}{KC}=\frac{PB}{PC}$

So, $\triangle CAB \sim \triangle CKP \Rightarrow \angle CAB= \angle CKP= 90^\circ$
Attachments
asas.png
asas.png (45.19KiB)Viewed 9558 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