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Old Kyiv MO Geometry
Kyiv City MO Juniors Round2 2010+ geometry
2019.9.31
2019.9.31
Part of
Kyiv City MO Juniors Round2 2010+ geometry
Problems
(1)
r_1+r_2+r_3=r, indradii (2019 Kyiv City MO Round2 9.3.1)
Source:
9/18/2020
A circle
k
k
k
of radius
r
r
r
is inscribed in
△
A
B
C
\vartriangle ABC
△
A
BC
, tangent to the circle
k
k
k
, which are parallel respectively to the sides
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
A
CA
C
A
intersect the other sides of
△
A
B
C
\vartriangle ABC
△
A
BC
at points
M
,
N
;
P
,
Q
M, N; P, Q
M
,
N
;
P
,
Q
and
L
,
T
L, T
L
,
T
(
P
,
T
∈
A
B
P, T \in AB
P
,
T
∈
A
B
,
L
,
N
∈
B
C
L, N \in BC
L
,
N
∈
BC
and
M
,
Q
∈
A
C
M, Q\in AC
M
,
Q
∈
A
C
). Denote by
r
1
,
r
2
,
r
3
r_1,r_2,r_3
r
1
,
r
2
,
r
3
the radii of inscribed circles in triangles
M
N
C
,
P
Q
A
MNC, PQA
MNC
,
PQ
A
and
L
T
B
LTB
L
TB
. Prove that
r
1
+
r
2
+
r
3
=
r
r_1+r_2+r_3=r
r
1
+
r
2
+
r
3
=
r
.
geometry
inradius
incircle