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Romanian Masters of Mathematics Collection
2018 Romanian Master of Mathematics Shortlist
G2
G2
Part of
2018 Romanian Master of Mathematics Shortlist
Problems
(1)
Two tangents meet on BC
Source: 2018 RMM Shortlist G2
2/21/2019
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle, and let
S
S
S
and
T
T
T
be the midpoints of the sides
B
C
BC
BC
and
C
A
CA
C
A
, respectively. Suppose
M
M
M
is the midpoint of the segment
S
T
ST
ST
and the circle
ω
\omega
ω
through
A
,
M
A, M
A
,
M
and
T
T
T
meets the line
A
B
AB
A
B
again at
N
N
N
. The tangents of
ω
\omega
ω
at
M
M
M
and
N
N
N
meet at
P
P
P
. Prove that
P
P
P
lies on
B
C
BC
BC
if and only if the triangle
A
B
C
ABC
A
BC
is isosceles with apex at
A
A
A
.Proposed by Reza Kumara, Indonesia
geometry