The lines r,s,t and the two γ circles

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The lines r,s,t and the two γ circles

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WakeUp

#1 h Nov 21, 2010, 7:26 PM • 2 Y

Y by Adventure10, Mango247

Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$ . A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$ . Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$ , at the smaller distance from $AD$ . Line $EA$ meets $\gamma_2$ in $F$ . Let $t$ be the tangent to $\gamma_2$ at $F$ .
$(a)$ Prove that $t$ is parallel to $AC$ .
$(b)$ Prove that the lines $r,s,t$ are concurrent.

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WakeUp

#2 h Nov 21, 2010, 7:33 PM • 2 Y

Y by Adventure10, Mango247

See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=6265

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jicko

#3 h Jan 16, 2020, 5:38 AM • 2 Y

Y by Adventure10, Mango247

FIRST OBSERVE THAT $\angle$ $CEA$ = $\angle$ $ABD$ = $\angle$ $AFD$ $therefore$ $EC$ $\parallel$ TO $FD$
AND $\angle$ $ECA$ = $\angle$ $AEX$ = $\angle$ $FAD$ = $\angle$ $DFX$ ( $X =s \cap t$ )
THEN WE CONCLUDE THAT $AC \parallel t$
LET $X*=CBD \cap t$ we observe $\angle ACB = \angle CX*F =\angle BEA = \angle BXF$ ( SINCE $\angle BEX+\angle BFX = 180 ^{\circ}$ )
THEN WE ARE DONE AS $X=X*$

This post has been edited 1 time. Last edited by jicko, Jan 16, 2020, 5:39 AM
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