IGO 2016 Elementary/4
 Thamim Zahin
 Posts: 98
 Joined: Wed Aug 03, 2016 5:42 pm
IGO 2016 Elementary/4
4. In a rightangled triangle $ABC (\angle A = 90)$, the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of $\angle C$, find all possible values for angles $\angle B$ and $\angle C$.
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.

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 Joined: Sun Dec 11, 2016 2:01 pm
Re: IGO 2016 Elementary/4
Isn't BC the diameter of the circumcircle?
 ahmedittihad
 Posts: 175
 Joined: Mon Mar 28, 2016 6:21 pm
Re: IGO 2016 Elementary/4
Yes. It's a very trivial observation. Try to solve the problem.
Frankly, my dear, I don't give a damn.
 Thamim Zahin
 Posts: 98
 Joined: Wed Aug 03, 2016 5:42 pm
Re: IGO 2016 Elementary/4
The fun thing(or cruel) about this problem is: You don't even have to do angle chasing in this problem. But when it is saying something about $90^o$, all we want is diameter. Through, use everything you learnt in class $6,7,8$. By that, you might solve the problem.
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.
 ahmedittihad
 Posts: 175
 Joined: Mon Mar 28, 2016 6:21 pm
 Thamim Zahin
 Posts: 98
 Joined: Wed Aug 03, 2016 5:42 pm
Re: IGO 2016 Elementary/4
It is a very nasty problem. By the way, thanks me if you read the solution. It was a hard time to $LateX$ all of them. It might look big, but, it is not very hard.
Case 1: $AC > AB$.
Claim 1: In $\triangle ABC$, $\angle LBK=\angle LKB$.
Proof: Let $X$ be the midpoint of $KB$. We know that, $LX \perp BK$. So. by $SAS$, $\triangle LXK \cong \triangle LXB$.
We denote $\angle LKB=\angle LBK=2a$. that means $\angle KLA=4a$.
Claim 2: $\angle KBC= 45^oa$
Proof: Like Claim 1, we can easily show that $\triangle KBC$ is isosceles and $KB=KC$. Now $\angle BKC= \angle KBA+\angle BAK= 90^o+2a$. So, $\angle KBC= \frac{180^o90^o2a}{2}=45a$.
Let $A'$ be the altitude on $BC$ from $L$.
Claim 3: $LA=LA'$
Proof: $CL=CL$, $\angle A'CL=\angle ACL$ and $\angle LAC=\angle LA'C$. So, by $ASA$, $\triangle ACL \cong \triangle A'CL$. So, $LA=LA'$.
Claim 4: $\angle LKA=\angle LBA'$.
Proof: $LA=LA'$[from claim 3],$LB=LK$[from claim 1], $\angle LAK=LA'B=90^o$. So, by $SSA$, $\triangle LKA \cong \triangle LA'B$. So,$\angle LKA=\angle LBA'$.
Now, $\angle LKA= 90^o4a$, $$\angle LBA=45^oa+21^o=45+a$$
$$\Rightarrow 90^o4a=45^o+a$$
$$\Rightarrow a=9^o$$.
So, $\angle B= 45^o+a=54^o$. So. $\angle C=90^o54^o=36^o$.
Case 2: $AC < AB$
Claim 5: $\angle LKB = \angle LBK$.
Proof: Same as Claim 1.
Let denote $\angle LKB = \angle LBK = a$. So, $\angle KLA=2a$.
Let $A'$ be the altitude of $L$ on $BC$.
Claim 6: $LA=LA'$
Proof: $CL=CL$, $\angle LCA=\angle LCA'$. $\angle CA'L=\angle CAL$. So, $\triangle CLA \cong \triangle CLA'$.So, $LA=LA'$.
Claim 7: $\angle AKL=\angle A'BL$.
Proof: $LK=LB$[Claim 5], $LA=LA'$[CLaim 6], $\angle LA'B=\angle LAK=90^o$. So, by $SSA$, $\triangle LAK \cong \triangle LA'B$.By that, $\angle AKL=\angle A'BL$.
Claim 8: $\triangle KBC$ is equilateral.
From [Claim 2] and [Claim 7], we got, $\angle LKB = \angle LBK$ and angle $AKL=\angle A'BL$. By combining them, $\angle CBK=\angle BKC$. Or, $CB=CK$. And we have, $KC=KB$[the perpendicular bisector $BC$ intersect the line $AC$ at $K$].
From that we get $CB=BK=KC$. Or, $\triangle KBC$ is equilateral.
That means. $\angle C=60^o$. So. $\angle B=90^o60^o=30^o$.
Case 3: $AC=AB$.
Claim 9: It is impossible.
In this case, $K \equiv A$ and $L$ is the midpoint of $AB$. Let $T$ be a point on $BC$ such that $LT\perp BC$. We know that the line $CL$ is the internal bisector of $\angle C$, so $LT = LA = LB$ which is impossible.
So, all possible solutions are: $(\angle B, \angle C)=(54^o,36^o), (30^o,60^o)$.
Case 1: $AC > AB$.
Claim 1: In $\triangle ABC$, $\angle LBK=\angle LKB$.
Proof: Let $X$ be the midpoint of $KB$. We know that, $LX \perp BK$. So. by $SAS$, $\triangle LXK \cong \triangle LXB$.
We denote $\angle LKB=\angle LBK=2a$. that means $\angle KLA=4a$.
Claim 2: $\angle KBC= 45^oa$
Proof: Like Claim 1, we can easily show that $\triangle KBC$ is isosceles and $KB=KC$. Now $\angle BKC= \angle KBA+\angle BAK= 90^o+2a$. So, $\angle KBC= \frac{180^o90^o2a}{2}=45a$.
Let $A'$ be the altitude on $BC$ from $L$.
Claim 3: $LA=LA'$
Proof: $CL=CL$, $\angle A'CL=\angle ACL$ and $\angle LAC=\angle LA'C$. So, by $ASA$, $\triangle ACL \cong \triangle A'CL$. So, $LA=LA'$.
Claim 4: $\angle LKA=\angle LBA'$.
Proof: $LA=LA'$[from claim 3],$LB=LK$[from claim 1], $\angle LAK=LA'B=90^o$. So, by $SSA$, $\triangle LKA \cong \triangle LA'B$. So,$\angle LKA=\angle LBA'$.
Now, $\angle LKA= 90^o4a$, $$\angle LBA=45^oa+21^o=45+a$$
$$\Rightarrow 90^o4a=45^o+a$$
$$\Rightarrow a=9^o$$.
So, $\angle B= 45^o+a=54^o$. So. $\angle C=90^o54^o=36^o$.
Case 2: $AC < AB$
Claim 5: $\angle LKB = \angle LBK$.
Proof: Same as Claim 1.
Let denote $\angle LKB = \angle LBK = a$. So, $\angle KLA=2a$.
Let $A'$ be the altitude of $L$ on $BC$.
Claim 6: $LA=LA'$
Proof: $CL=CL$, $\angle LCA=\angle LCA'$. $\angle CA'L=\angle CAL$. So, $\triangle CLA \cong \triangle CLA'$.So, $LA=LA'$.
Claim 7: $\angle AKL=\angle A'BL$.
Proof: $LK=LB$[Claim 5], $LA=LA'$[CLaim 6], $\angle LA'B=\angle LAK=90^o$. So, by $SSA$, $\triangle LAK \cong \triangle LA'B$.By that, $\angle AKL=\angle A'BL$.
Claim 8: $\triangle KBC$ is equilateral.
From [Claim 2] and [Claim 7], we got, $\angle LKB = \angle LBK$ and angle $AKL=\angle A'BL$. By combining them, $\angle CBK=\angle BKC$. Or, $CB=CK$. And we have, $KC=KB$[the perpendicular bisector $BC$ intersect the line $AC$ at $K$].
From that we get $CB=BK=KC$. Or, $\triangle KBC$ is equilateral.
That means. $\angle C=60^o$. So. $\angle B=90^o60^o=30^o$.
Case 3: $AC=AB$.
Claim 9: It is impossible.
In this case, $K \equiv A$ and $L$ is the midpoint of $AB$. Let $T$ be a point on $BC$ such that $LT\perp BC$. We know that the line $CL$ is the internal bisector of $\angle C$, so $LT = LA = LB$ which is impossible.
So, all possible solutions are: $(\angle B, \angle C)=(54^o,36^o), (30^o,60^o)$.
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Last edited by Thamim Zahin on Thu Mar 02, 2017 12:51 am, edited 1 time in total.
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 Elementary/4
DONE. Didn't use any angle chasing as I saidahmedittihad wrote:Why couldn't you? Thamim
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 Elementary/4
I more solution Case 1:$AC>AB$
By a well know lemma we know that $BLKC$ is cyclic.
Now denote, $\angle LBK=\angle LCK=\angle LCB=\angle LKB=a$.
And we have that $KC=KB$. So, $\angle KBC=2a$. So, we have $a+2a+a+a=90^o \Rightarrow a=18^o$.
That means $2a=\angle C=18^o\times 2=36^o$. So, $\angle B= 90^o36^o=54^o$.
By a well know lemma we know that $BLKC$ is cyclic.
Now denote, $\angle LBK=\angle LCK=\angle LCB=\angle LKB=a$.
And we have that $KC=KB$. So, $\angle KBC=2a$. So, we have $a+2a+a+a=90^o \Rightarrow a=18^o$.
That means $2a=\angle C=18^o\times 2=36^o$. So, $\angle B= 90^o36^o=54^o$.
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.

 Posts: 50
 Joined: Sun Dec 11, 2016 2:01 pm
Re: IGO 2016 Elementary/4
It is just the official solution....is there any unique technique to solve that?