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Problems
Contests
National and Regional Contests
Taiwan Contests
TST Round 3
2018 Taiwan TST Round 3
3
3
Part of
2018 Taiwan TST Round 3
Problems
(1)
Right angles
Source: 2018 Taiwan TST Round 3
4/2/2020
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
, and
ℓ
\ell
ℓ
be the perpendicular bisector of
A
I
AI
A
I
. Suppose that
P
P
P
is on the circumcircle of triangle
A
B
C
ABC
A
BC
, and line
A
P
AP
A
P
and
ℓ
\ell
ℓ
intersect at point
Q
Q
Q
. Point
R
R
R
is on
ℓ
\ell
ℓ
such that
∠
I
P
R
=
9
0
∘
\angle IPR = 90^{\circ}
∠
I
PR
=
9
0
∘
.Suppose that line
I
Q
IQ
I
Q
and the midsegment of
A
B
C
ABC
A
BC
that is parallel to
B
C
BC
BC
intersect at
M
M
M
. Show that
∠
A
M
R
=
9
0
∘
\angle AMR = 90^{\circ}
∠
A
MR
=
9
0
∘
(Note: In a triangle, a line connecting two midpoints is called a midsegment.)
geometry
Taiwan