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Flanders Math Olympiad
1991 Flanders Math Olympiad
3
triangle (ratios)
triangle (ratios)
Source: flanders '91
September 27, 2005
ratio
geometry
trigonometry
Problem Statement
Given
Δ
A
B
C
\Delta ABC
Δ
A
BC
equilateral, with
X
∈
[
A
,
B
]
X\in[A,B]
X
∈
[
A
,
B
]
. Then we define unique points Y,Z so that
Y
∈
[
B
,
C
]
Y\in[B,C]
Y
∈
[
B
,
C
]
,
Z
∈
[
A
,
C
]
Z\in[A,C]
Z
∈
[
A
,
C
]
,
Δ
X
Y
Z
\Delta XYZ
Δ
X
Y
Z
equilateral. If
A
r
e
a
(
Δ
A
B
C
)
=
2
⋅
A
r
e
a
(
Δ
X
Y
Z
)
Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)
A
re
a
(
Δ
A
BC
)
=
2
⋅
A
re
a
(
Δ
X
Y
Z
)
, find the ratio of
A
X
X
B
,
B
Y
Y
C
,
C
Z
Z
A
\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}
XB
A
X
,
Y
C
B
Y
,
Z
A
CZ
.
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