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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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jlacosta
Apr 2, 2025
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N.S. condition of passing a fixed point for a function
Kunihiko_Chikaya   1
N 28 minutes ago by Mathzeus1024
Let $ f(t)$ be a function defined in any real numbers $ t$ with $ f(0)\neq 0.$ Prove that on the $ x-y$ plane, the line $ l_t : tx+f(t) y=1$ passes through the fixed point which isn't on the $ y$ axis in regardless of the value of $ t$ if only if $ f(t)$ is a linear function in $ t$.
1 reply
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Kunihiko_Chikaya
Sep 6, 2009
Mathzeus1024
28 minutes ago
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Dot product
SomeonecoolLovesMaths   4
N 31 minutes ago by quasar_lord
How to prove that dot product is distributive?
4 replies
SomeonecoolLovesMaths
Yesterday at 6:06 PM
quasar_lord
31 minutes ago
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Geometry Angle Chasing
Sid-darth-vater   5
N 2 hours ago by sunken rock
Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!
5 replies
Sid-darth-vater
Apr 21, 2025
sunken rock
2 hours ago
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About my new website
Samujjal101   21
N 2 hours ago by Samujjal101
Hi everybody!
I'm registering some of the finest minds in math into my website.. it's not completely developed.. but still if you want we would be very grateful to have you!
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Maths-matchmaker is a website for connecting math minds together with a mission to unite together. It uses a matching algorithm to match 1:1 with like minded peers based on their interests or topics in math
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Samujjal101
Yesterday at 2:28 PM
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2 equilaterals on sides of a square, areas compare
parmenides51   5
N Jun 8, 2024 by parmenides51
$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle.
IMAGE
5 replies
parmenides51
May 11, 2019
parmenides51
Jun 8, 2024
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2 equilaterals on sides of a square, areas compare
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Reveal topic tags
geometry
area of a triangle
Equilateral
square
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parmenides51
30650 posts
parmenides51
#1 h  May 11, 2019, 2:01 AM • 1 Y
Y by Adventure10
$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle.
https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif
This post has been edited 2 times. Last edited by parmenides51, Aug 19, 2019, 9:31 PM
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khanhnx
1618 posts
khanhnx
#2 h  May 11, 2019, 4:00 AM • 2 Y
Y by Adventure10, Mango247
Denote: $S_P$ is area of polygon $P$
We have: $S_{FED} = \dfrac{ED . FD . \sin \widehat{EDF}}{2} = \dfrac{AB^2}{4} = \dfrac{S_{ABCD}}{4} = S_{DCO}$
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zuss77
520 posts
zuss77
#3 h  May 11, 2019, 6:48 AM • 5 Y
Y by sunken rock, Adventure10, Mango247, Mango247, Mango247
Just notice that $\triangle FDC = \triangle FDE$ and with $\triangle DCO$ it has common base $DC$ and equal height.
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jayme
9787 posts
jayme
#4 h  Feb 17, 2020, 3:02 PM • 1 Y
Y by Adventure10
Dear Mathlinkers,

http://jl.ayme.pagesperso-orange.fr/Docs/Miniatures%20Geometriques%20addendum%20VII.pdf p. 69...

Sincerely
Jean-Louis
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SansPapyrus683
460 posts
SansPapyrus683
#5 h  Feb 17, 2020, 4:34 PM • 1 Y
Y by Adventure10
@above we don't speak french
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parmenides51
30650 posts
parmenides51
#6 h  Jun 8, 2024, 3:31 PM
Y by
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This post has been edited 2 times. Last edited by parmenides51, Jun 21, 2024, 8:40 AM
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