Greece NMO 2013#P4

[Phlembac Adib Hasan]

Let a triangle $ABC$ inscribed in circle $c(O,R)$ and $D$ an arbitrary point on $BC$(different from the midpoint).The circumscribed circle of $BOD$,which is $(c_1)$, meets $c(O,R)$ at $K$ and $AB$ at $Z$.The circumscribed circle of $COD$ $(c_2)$,meets $c(O,R)$ at $M$ and $AC$ at $E$.Finally, the circumscribed circle of $AEZ$ $(c_3)$,meets $c(O,R)$ at $N$.Prove that $\triangle{ABC}\cong \triangle{KMN}.$

(Juniors, try this one. Such contest problems are always worth trying, they give you a new level of confidence-even if you can't solve them!)

[Thanic Nur Samin]

$\Delta ABC$ is inscribed in $c$
=>$c$ is the circumcircle of $\Delta ABC$
=>$c$ crosses the points $A,B,C$.

Again,the circumcircles $c_1,c_2,c_3$ intersect $\Delta BOD$,$\Delta COD$,$\Delta AEZ$ in B,C and A [Circumcircles intersect the points of the triangle]
But c crosses the points too.So they intersect with each other in B,C,A.Which means B&K,C&N,A&M are the same.
Obviously, $\Delta ABC \cong \Delta KMN$

[*Mahi*]

Again,the circumcircles $c_1,c_2,c_3$ intersect $\Delta BOD$,$\Delta COD$,$\Delta AEZ$ in B,C and A [Circumcircles intersect the points of the triangle]
But c crosses the points too.So they intersect with each other in B,C,A.

Two intersecting circles may have two points in common, not just one.