212. A simple applications of the Casey's theorem.
by Virgil Nicula, Jan 22, 2011, 7:54 AM
Proposed problem 1. Let 
be an isosceles and 
-right triangle with the circumcircle 
. Consider a circle 
which is
tangent to
and interior tangent to so that separates 
, . Denote 
for which 
. Prove that 
.
Proof. Denote
and 
, 
and 
. Observe that 
. But we also note that since 
, we have 
.
An easy extension. Let be a triangle with the circumcircle 
. Consider a circle which is tangent to 
in 
and interior tangent to so that separates , . Denote for which . Prove that 
.
Proof. I"ll use the Casey's theorem to the circles , 
, 
and 
which are interior tangent to the circle . Therefore, 
.
Particular case. Let be an isosceles triangle with the circumcircle . Consider a circle which is tangent to
and interior tangent to so that separates , . Denote for which . Prove that .
Proof. Denote and . From the well-known property obtain that 
.
Thus,
.
PP2. Let be a triangle with the incircle 
and the circumcircle . Consider the circle 
which is tangent to 
, 
and which
is internal tangent to . Denote 
. The tangent from 
to touches in 
. Prove that 
.
Proof. Assume w.l.o.g. that
and let 
be the tangency points of 
with 
By Casey's theorem for , , ,
all tangent to
, we get 
If 
is the reflection of
across
then 
. Hence 
. Using the Ptolemy's theorem
for
we deduce that 
. Together with the expression 
we obtain 
.
PP3. Let , 
be two circles which are interior tangent to the circle in the points , respectively and let 
be two points so that is the common interior
tangent in of , and the line separates , 
. Denote the exterior tangent 
of the circles , so that 
and 
. Prove that 
.
Proof. Apply the Casey's theorem to
and the degenerate circles (with the null radii) 
.
PP4. Let
with the circumcircle . Let the circle 
which is internal tangent to and is tangent to in 
. Let the circle
which is external tangent to and is tangent to in 
. Prove that 
, where 
.
Proof. Denote
. Apply the Casey's theorem to the degerate circles 
and the circles 
.
be an isosceles and 
-right triangle with the circumcircle 
. Consider a circle 
which istangent to
and interior tangent to so that separates 
, . Denote 
for which 
. Prove that 
.Proof. Denote
and 
, 
and 
. Observe that 
. But we also note that since 
, we have 
.An easy extension. Let
be a triangle with the circumcircle 
. Consider a circle which is tangent to 
in 
and interior tangent to
so that separates , . Denote for which . Prove that 
. Proof. I"ll use the Casey's theorem to the circles
, 
, 
and 
which are interior tangent to the circle . Therefore, 
. Particular case. Let
be an isosceles triangle with the circumcircle . Consider a circle which is tangent to and interior tangent to so that separates , . Denote for which . Prove that .Proof. Denote
and . From the well-known property obtain that 
.Thus,
.PP2. Let
be a triangle with the incircle 
and the circumcircle . Consider the circle 
which is tangent to 
, 
and whichis internal tangent to
. Denote 
. The tangent from 
to touches in 
. Prove that 
.Proof. Assume w.l.o.g. that
and let 
be the tangency points of 
with 
By Casey's theorem for , , , all tangent to
, we get 
If 
is the reflection of across
then 
. Hence 
. Using the Ptolemy's theoremfor
we deduce that 
. Together with the expression 
we obtain 
.PP3. Let
, 
be two circles which are interior tangent to the circle in the points , respectively and let 
be two points so that is the common interiortangent in
of , and the line separates , 
. Denote the exterior tangent 
of the circles , so that 
and 
. Prove that 
.Proof. Apply the Casey's theorem to
and the degenerate circles (with the null radii) 
.PP4. Let
with the circumcircle . Let the circle 
which is internal tangent to and is tangent to in 
. Let the circle
which is external tangent to and is tangent to in 
. Prove that 
, where 
.Proof. Denote
. Apply the Casey's theorem to the degerate circles 
and the circles 
.This post has been edited 28 times. Last edited by Virgil Nicula, Nov 26, 2015, 6:19 PM
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