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National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2007 Junior Balkan Team Selection Tests - Moldova
3
3
Part of
2007 Junior Balkan Team Selection Tests - Moldova
Problems
(1)
incenter wanted, 1/AE + 1/AF =( a + b + c)/ bc.
Source: 2007 Moldova JBMO TST 1.3
7/12/2020
Let
A
B
C
ABC
A
BC
be a triangle with
B
C
=
a
,
A
C
=
b
BC = a, AC = b
BC
=
a
,
A
C
=
b
and
A
B
=
c
AB = c
A
B
=
c
. A point
P
P
P
inside the triangle has the property that for any line passing through
P
P
P
and intersects the lines
A
B
AB
A
B
and
A
C
AC
A
C
in the distinct points
E
E
E
and
F
F
F
we have the relation
1
A
E
+
1
A
F
=
a
+
b
+
c
b
c
\frac{1}{AE} +\frac{1}{AF} =\frac{a + b + c}{bc}
A
E
1
+
A
F
1
=
b
c
a
+
b
+
c
. Prove that the point
P
P
P
is the center of the circle inscribed in the triangle
A
B
C
ABC
A
BC
.
geometry
incenter