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Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
12
12
Part of
2019 Sharygin Geometry Olympiad
Problems
(1)
Max of min of cevian segments
Source: Sharygin CR 2019 P12 (Grade 8 - 11)
3/6/2019
Let
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
be an acute-angled triangle inscribed into a unit circle centered at
O
O
O
. The cevians from
A
i
A_i
A
i
passing through
O
O
O
meet the opposite sides at points
B
i
B_i
B
i
(
i
=
1
,
2
,
3
)
(i = 1, 2, 3)
(
i
=
1
,
2
,
3
)
respectively.[*] Find the minimal possible length of the longest of three segments
B
i
O
B_iO
B
i
O
. [*] Find the maximal possible length of the shortest of three segments
B
i
O
B_iO
B
i
O
.
geometry