2000 IMO Problems/Problem 1

Revision as of 12:43, 13 August 2022 by Sayemsub15 (talk | contribs) (Solution)


Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$. Let $AB$ be the line tangent to these circles at $A$ and $B$, respectively, so that $M$ lies closer to $AB$ than $N$. Let $CD$ be the line parallel to $AB$ and passing through the point $M$, with $C$ on $G_1$ and $D$ on $G_2$. Lines $AC$ and $BD$ meet at $E$; lines $AN$ and $CD$ meet at $P$; lines $BN$ and $CD$ meet at $Q$. Show that $EP=EQ$.


$\textbf{Lemma:}$ Given a triangle, $ABC$ and a point $P$ in it's interior, assume that the circumcircles of $\triangle{ACP}$ and    $\triangle{ABP}$ are tangent to $BC$. Prove that ray $AP$ bisects $BC$.
$\textbf{Proof:}$ Let the intersection of $AP$ and $BC$ be $D$. By power of a point, $BD^2=AD(PD)$ and $CD^2=AD(PD)$, so $BD=CD$.

$\textbf{Proof of problem:}$ Let ray $NM$ intersect $AB$ at $X$. By our lemma, $\textit{(the two circles are tangent to AB)}$, $X$ bisects $AB$. Since $\triangle{NAX}$ and $\triangle{NPM}$ are similar, and $\triangle{NBX}$ and $\triangle{NQM}$ are similar implies $M$ bisects $PQ$.

By simple parallel line rules, $\angle{ABM}=\angle{EBA}$. Similarly, $\angle{BAM}=\angle{EAB}$, so by $\textit{ASA}$ criterion, $\triangle{ABM}$ and $\triangle{EAB}$ are congruent.

Now, $\angle{EBA} = \angle{ABM} = \angle{BDM}$ and $\angle{ABM} = \angle{BMD}$ since $CD$ is parallel to $AB$. But $AB$ is tangent to the circumcircle of $\triangle{BMD}$ hence $\angle{ABM} = \angle{BDM}$ and that implies $\angle{BMD} = \angle{BDM} .$So$\triangle{BMD}$ is isosceles and $\angle{BMD}=\angle{BDM}$ .

Join points $E$ and $M$, $EM$ is perpendicular on $PQ$ (why?), previously we proved $MP = MQ$, hence $\triangle{EPQ}$ is isoscles and $EP = EQ$ .