Circles
and
with centres
and
are externally tangent at point
and internally tangent to a circle
at points
and
respectively. Line
is the common tangent of
and
at
. Let
be the diameter of
perpendicular to
, so that
are on the same side of
. Prove that lines
,
,
and
are concurrent.
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Circles $w_{1}$ and $w_{2}$ with centres $O_{1}$ and $O_{2}$ are externally tangent at point $D$ and internally tangent to a circle $w$ at points $E$ and $F$ respectively. Line $t$ is the common tangent of $w_{1}$ and $w_{2}$ at $D$. Let $AB$ be the diameter of $w$ perpendicular to $t$, so that $A, E, O_{1}$ are on the same side of $t$. Prove that lines $AO_{1}$, $BO_{2}$, $EF$ and $t$ are concurrent.
Školjka