AoPS - Problem

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Problem

Source: Romanian JBTST IV 2007, problem 2

Tags: geometry, power of a point, radical axis, geometry proposed

pohoatza

13.05.2007 14:45

Let $w_{1}$ and $w_{2}$ be two circles which intersect at points $A$ and $B$ . Consider $w_{3}$ another circle which cuts $w_{1}$ in $D,E$ , and it is tangent to $w_{2}$ in the point $C$ , and also tangent to $AB$ in $F$ . Consider $G \in DE \cap AB$ , and $H$ the symetric point of $F$ w.r.t $G$ . Find $\angle{HCF}$ .

pontios

13.05.2007 15:29

$G$ is on the radical axis of

$w_{1},w_{2}$

$G$ is on the radical axis of

$w_{1},w_{3}$ So, the line

$GC$ is the radical axis of

$w_{2},w_{3}$ . As a result,

$GC$ is tangent to

$w_{3}$ . So

$GC=GF$ . But

$G$ is the midpoint of

$FH$ . So,

$\triangle HCF$ is right at

$C$

lasha.p

30.11.2009 23:23

radical line of w1 and w2 is AB,radical line of w1 and w3 is DE and w2 and w3 are tangent in C,so CG is olso radical line of w2 and w3 => GF=GC=GH => <HCF=90