IGO 2016 Medium/4
- Thamim Zahin
- Posts:98
- Joined:Wed Aug 03, 2016 5:42 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.
- Thamim Zahin
- Posts:98
- Joined:Wed Aug 03, 2016 5:42 pm
Re: IGO 2016 Medium/4
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, $\triangle CAB \sim \triangle CKP \Rightarrow \angle CAB= \angle CKP= 90^\circ$
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 (45.19KiB)Viewed 9863 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.