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National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2014 NZMOC Camp Selection Problems
9
9
Part of
2014 NZMOC Camp Selection Problems
Problems
(1)
AX x BY = AI x BI, circle with center a midpoint, triangle with that incircle
Source: 2014 NZOMC Camp Selections p9
1/10/2021
Let
A
B
AB
A
B
be a line segment with midpoint
I
I
I
. A circle, centred at
I
I
I
has diameter less than the length of the segment. A triangle
A
B
C
ABC
A
BC
is tangent to the circle on sides
A
C
AC
A
C
and
B
C
BC
BC
. On
A
C
AC
A
C
a point
X
X
X
is given, and on
B
C
BC
BC
a point
Y
Y
Y
is given such that
X
Y
XY
X
Y
is also tangent to the circle (in particular
X
X
X
lies between the point of tangency of the circle with
A
C
AC
A
C
and
C
C
C
, and similarly
Y
Y
Y
lies between the point of tangency of the circle with
B
C
BC
BC
and
C
C
C
. Prove that
A
X
⋅
B
Y
=
A
I
⋅
B
I
AX \cdot BY = AI \cdot BI
A
X
⋅
B
Y
=
A
I
⋅
B
I
.
geometry
incircle