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Adygea Teachers' Geometry Olympiad
2023 Adygea Teachers' Geometry Olympiad
2023 Adygea Teachers' Geometry Olympiad
Part of
Adygea Teachers' Geometry Olympiad
Subcontests
(3)
4
1
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cevians in equilateral (2023 Adygea Teachers' Geometry Olympiad p4)
In the equilateral triangle
A
B
C
ABC
A
BC
(
A
B
=
2
AB = 2
A
B
=
2
), cevians are drawn that do not intersect at one point. It turned out that the pairwise intersection points of these cevians lie on the inscribed circle of triangle
A
B
C
ABC
A
BC
. Find the length of the cevian segment from the vertex of the triangle to the nearest point of intersection with the circle.
3
1
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areas by 3 cevians (2023 Adygea Teachers' Geometry Olympiad p3)
Three cevians are drawn in a triangle that do not intersect at one point. In this case,
4
4
4
triangles and
3
3
3
quadrangles were formed. Find the sum of the areas of the quadrilaterals if the area of each of the four triangles is
8
8
8
.
1-2
1
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areas by cevians (2023 Adygea Teachers' Geometry Olympiad p1-2)
Three cevians divided the triangle into six triangles, the areas of which are marked in the figure. 1) Prove that
S
1
⋅
S
2
⋅
S
3
=
Q
1
⋅
Q
2
⋅
Q
3
S_1 \cdot S_2 \cdot S_3 =Q_1 \cdot Q_2 \cdot Q_3
S
1
⋅
S
2
⋅
S
3
=
Q
1
⋅
Q
2
⋅
Q
3
.2) Determine whether it is true that if
S
1
=
S
2
=
S
3
S_1 = S_2 = S_3
S
1
=
S
2
=
S
3
, then
Q
1
=
Q
2
=
Q
3
Q_1 = Q_2 = Q_3
Q
1
=
Q
2
=
Q
3
.https://cdn.artofproblemsolving.com/attachments/c/d/3e847223b24f783551373e612283e10e477e62.png