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Olympiad Geometry problem-second time posting
kjhgyuio   5
N an hour ago by kjhgyuio
Source: smo problem
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
5 replies
kjhgyuio
Today at 1:03 AM
kjhgyuio
an hour ago
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Locus of a point on the side of a square
EmersonSoriano   0
an hour ago
Source: 2018 Peru TST Cono Sur P7
Let $ABCD$ be a fixed square and $K$ a variable point on segment $AD$. The square $KLMN$ is constructed such that $B$ is on segment $LM$ and $C$ is on segment $MN$. Let $T$ be the intersection point of lines $LA$ and $ND$. Find the locus of $T$ as $K$ varies along segment $AD$.
0 replies
EmersonSoriano
an hour ago
0 replies
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calculate the perimeter of triangle MNP
PennyLane_31   1
N 2 hours ago by TheBaiano
Source: 2024 5th OMpD L2 P2 - Brazil - Olimpíada Matemáticos por Diversão
Let $ABCD$ be a convex quadrilateral, and $M$, $N$, and $P$ be the midpoints of diagonals $AC$ and $BD$, and side $AD$, respectively. Also, suppose that $\angle{ABC} + \angle{DCB} = 90$ and that $AB = 6$, $CD = 8$. Calculate the perimeter of triangle $MNP$.
1 reply
PennyLane_31
Oct 16, 2024
TheBaiano
2 hours ago
J
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Geo Final but hard to solve with Conics...
Seungjun_Lee   5
N 3 hours ago by L13832
Source: 2025 Korea Winter Program Practice Test P4
Let $\omega$ be the circumcircle of triangle $ABC$ with center $O$, and the $A$ inmixtilinear circle is tangent to $AB, AC, \omega$ at $D,E,T$ respectively. $P$ is the intersection of $TO$ and $DE$ and $X$ is the intersection of $AP$ and $\omega$. Prove that the isogonal conjugate of $P$ lies on the line passing through the midpoint of $BC$ and $X$.
5 replies
Seungjun_Lee
Jan 18, 2025
L13832
3 hours ago
J
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April Fools Geometry
awesomeming327.   5
N Today at 4:31 PM by PEKKA
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ be the projection from $A$ onto $BC$. Let $E$ be a point on the extension of $AD$ past $D$ such that $\angle BAC+\angle BEC=90^\circ$. Let $L$ be on the perpendicular bisector of $AE$ such that $L$ and $C$ are on the same side of $AE$ and
\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]Let the reflection of $D$ across $AB$ and $AC$ be $W$ and $Y$, respectively. Let $X\in AW$ and $Z\in AY$ such that $\angle XBE=\angle ZCE=90^\circ$. Let $EX$ and $EZ$ intersect the circumcircles of $EBD$ and $ECD$ at $J$ and $K$, respectively. Let $LB$ and $LC$ intersect $WJ$ and $YK$ at $P$ and $Q$. Let $PQ$ intersect $BC$ at $F$. Prove that $FB/FC=DB/DC$.
5 replies
awesomeming327.
Yesterday at 2:52 PM
PEKKA
Today at 4:31 PM
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Assisted perpendicular chasing
sarjinius   3
N Today at 3:59 PM by ZeroHero
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
3 replies
sarjinius
Mar 9, 2025
ZeroHero
Today at 3:59 PM
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IMOC 2017 G2 , (ABC) <= (DEF) . perpendiculars related
parmenides51   7
N Today at 2:15 PM by AshAuktober
Source: https://artofproblemsolving.com/community/c6h1740077p11309077
Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$
7 replies
parmenides51
Mar 20, 2020
AshAuktober
Today at 2:15 PM
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Show that AB/AC=BF/FC
syk0526   75
N Today at 2:04 PM by AshAuktober
Source: APMO 2012 #4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.

(Here we denote $XY$ the length of the line segment $XY$.)
75 replies
syk0526
Apr 2, 2012
AshAuktober
Today at 2:04 PM
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Sequel to IMO 2016/1
Scilyse   6
N Today at 12:35 PM by L13832
Source: 2024 MODS Geometry Contest, Problem 4 of 6
Let $ABCD$ be a parallelogram. Let line $\ell$ externally bisect $\angle DCA$ and let $\ell'$ be the line passing through $D$ which is parallel to line $AC$. Suppose that $\ell'$ meets line $AB$ at point $E$ and $\ell$ at point $F$, and that $\ell$ meets the internal bisector of $\angle BAC$ at point $X$. Further let circle $EXF$ meet line $BX$ at point $Y \neq X$ and the internal bisector of $\angle DCA$ meet circle $AXC$ at point $Z \neq C$.
Prove that points $D$, $X$, $Y$, and $Z$ are concyclic.

Proposed by squarc_rs3v2m
6 replies
Scilyse
Mar 15, 2024
L13832
Today at 12:35 PM
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Vector geometry with unusual points
Ciobi_   0
Today at 12:28 PM
Source: Romania NMO 2025 9.2
Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$, circumradius $R$ and orthocenter $H$. Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$. Define $B_1$ and $C_1$ similarly.
If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that $\triangle ABC$ is equilateral.
0 replies
Ciobi_
Today at 12:28 PM
0 replies
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