Found problems: 6
2011 Flanders Math Olympiad, 4
Given is a triangle $ABC$ and points $D$ and $E$, respectively on $] BC [$ and $] AB [$. $F$ it is intersection of lines $AD$ and $CE$. We denote as $| CD | = a, | BD | = b, | DF | = c$ and $| AF | = d$. Determine the ratio $\frac{| BE |}{|AE |}$ in terms of $a, b, c$ and $d$
[img]https://cdn.artofproblemsolving.com/attachments/5/7/856c97045db2d9a26841ad00996a2b809addaa.png[/img]
2021 Sharygin Geometry Olympiad, 9.1
Three cevians concur at a point lying inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form (in some order) a geometric progression. Prove that the lengths of the cevians also form a geometric progression.
2012 Sharygin Geometry Olympiad, 2
We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal.
(A.Zaslavsky, B.Frenkin)
2006 Sharygin Geometry Olympiad, 21
On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken.
Prove that for the areas of the corresponding triangles, the inequality holds:
$$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$
and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.
2012 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an arbitrary triangle, and let $M, N, P$ be any three points on the sides $BC, CA, AB$ such that the lines $AM, BN, CP$ concur. Let the parallel to the line $AB$ through the point $N$ meet the line $MP$ at a point $E$, and let the parallel to the line $AB$ through the point $M$ meet the line $NP$ at a point $F$. Then, the lines $CP, MN$ and $EF$ are concurrent.
[hide=MOP 97 problem]Let $ABC$ be a triangle, and $M$, $N$, $P$ the points where its incircle touches the sides $BC$, $CA$, $AB$, respectively. The parallel to $AB$ through $N$ meets $MP$ at $E$, and the parallel to $AB$ through $M$ meets $NP$ at $F$. Prove that the lines $CP$, $MN$, $EF$ are concurrent. [url=https://artofproblemsolving.com/community/c6h22324p143462]also[/url][/hide]
2017 Sharygin Geometry Olympiad, P19
Let cevians $AA', BB'$ and $CC'$ of triangle $ABC$ concur at point $P.$ The circumcircle of triangle $PA'B'$ meets $AC$ and $BC$ at points $M$ and $N$ respectively, and the circumcircles of triangles $PC'B'$ and $PA'C'$ meet $AC$ and $BC$ for the second time respectively at points $K$ and $L$. The line $c$ passes through the midpoints of segments $MN$ and $KL$. The lines $a$ and $b$ are defined similarly. Prove that $a$, $b$ and $c$ concur.