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2023 Assam Mathematics Olympiad
17
Assam Mathematics Olympiad 2023 Category III Q17
Assam Mathematics Olympiad 2023 Category III Q17
Source:
September 11, 2024
geometry
circumcircle
Problem Statement
If in
△
A
B
C
\bigtriangleup ABC
△
A
BC
,
A
D
AD
A
D
is the altitude and
A
E
AE
A
E
is the diameter of the circumcircle through
A
A
A
, then prove that
A
B
⋅
A
C
=
A
D
⋅
A
E
AB\cdot AC = AD \cdot AE
A
B
⋅
A
C
=
A
D
⋅
A
E
. Use this result to show that if
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral then show that
A
C
⋅
(
A
B
⋅
B
C
+
C
D
⋅
D
A
)
=
B
D
⋅
(
D
A
⋅
A
B
+
B
C
⋅
C
D
)
AC \cdot (AB \cdot BC + CD \cdot DA) = BD\cdot (DA\cdot AB + BC \cdot CD)
A
C
⋅
(
A
B
⋅
BC
+
C
D
⋅
D
A
)
=
B
D
⋅
(
D
A
⋅
A
B
+
BC
⋅
C
D
)
.
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