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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1993 Moldova Team Selection Test
5
5
Part of
1993 Moldova Team Selection Test
Problems
(1)
Inside the acute triangle $ABC$ the point $P{}$ in on height $AA_1{}$.
Source: Moldova TST 1993
8/8/2023
Inside the acute triangle
A
B
C
ABC
A
BC
the point
P
P{}
P
in on height
A
A
1
AA_1{}
A
A
1
. Lines
B
P
BP{}
BP
and
C
P
CP{}
CP
intersect the sides
A
C
AC{}
A
C
and
A
B
AB{}
A
B
, respectively, in points
B
1
B_1{}
B
1
and
C
1
C_1{}
C
1
. Prove that: a)
A
A
1
AA_1{}
A
A
1
is the bisector of the angle
B
1
A
1
C
1
;
B_1A_1C_1;
B
1
A
1
C
1
;
b) if the lines
B
C
BC
BC
and
B
1
C
1
B_1C_1
B
1
C
1
are concurrent, then the position of theri intersection does not depend on
P
.
P.
P
.
geometry