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KoMaL A Problems
KoMaL A Problems 2018/2019
A. 750
A. 750
Part of
KoMaL A Problems 2018/2019
Problems
(1)
So Many Circles
Source: KöMaL A. 750
3/19/2022
Let
k
1
,
k
2
,
…
,
k
5
k_1,k_2,\ldots,k_5
k
1
,
k
2
,
…
,
k
5
be five circles in the lane such that
k
1
k_1
k
1
and
k
2
k_2
k
2
are externally tangent to each other at point
T
,
T,
T
,
k
3
k_3
k
3
and
k
4
k_4
k
4
are exetrnally tangent to both
k
1
k_1
k
1
and
k
2
,
k_2,
k
2
,
k
5
k_5
k
5
is externally tangent to
k
3
k_3
k
3
and
k
4
k_4
k
4
at points
U
U
U
and
V
,
V,
V
,
respectively, and
k
5
k_5
k
5
intersects
k
1
k_1
k
1
at
P
P
P
and
Q
,
Q,
Q
,
like shown in the figure. Prove that
P
U
Q
U
⋅
P
V
Q
V
=
P
T
2
Q
T
2
.
\frac{PU}{QU}\cdot\frac{PV}{QV}=\frac{PT^2}{QT^2}.
Q
U
P
U
⋅
Q
V
P
V
=
Q
T
2
P
T
2
.
geometry
komal
circles