nice geo

by Melid, Apr 23, 2025, 3:01 PM

Let ABCD be a cyclic quadrilateral, which is AB=7 and BC=6. Let E be a point on segment CD so that BE=9. Line BE and AD intersect at F. Suppose that A, D, and F lie in order. If AF=11 and DF:DE=7:6, find the length of segment CD.

geometry

Tangents forms triangle with two times less area

by NO_SQUARES, Apr 23, 2025, 9:08 AM

Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$ . Prove that $S_{DEF}=2S_{ABC}$ . ( $S_T$ is area of triangle $T$ ).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov

Attachments:

geometry

conics

parabola

Concurrency

by Dadgarnia, Mar 12, 2020, 10:54 AM

Let $ABC$ be an isosceles triangle ( $AB=AC$ ) with incenter $I$ . Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$ . $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$ , respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$ . Prove that $AD$ , $MN$ and $BC$ are concurrent.

Proposed by Alireza Dadgarnia

PAMO Problem 4: Perpendicular lines

by DylanN, Apr 9, 2019, 6:54 AM

The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$ . The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$ . Show that $AO$ is perpendicular to $EF$ .

geometry

PAMO

circumcircle

IMO 2014 Problem 4

by ipaper, Jul 9, 2014, 11:38 AM

Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$ . Let $M$ and $N$ be the points on $AP$ and $AQ$ , respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$ . Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$ .

Proposed by Giorgi Arabidze, Georgia.

barycentric coordinates

geometry solved

IMO 2012/5 Mockup

by v_Enhance, Jul 30, 2013, 5:19 AM

Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$ , and let $D$ be the foot of the altitude from $C$ . Let $X$ be a point in the interior of the segment $CD$ . Let $K$ be the point on the segment $AX$ such that $BK = BC$ . Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$ . The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$ ). Prove that $\angle ACT = \angle BCT$ .

This post has been edited 1 time. Last edited by v_Enhance, May 7, 2015, 1:33 AM
Reason: 90\dg should be 90^{\circ}

IMO Shortlist 2011, G4

by WakeUp, Jul 13, 2012, 11:41 AM

Let $ABC$ be an acute triangle with circumcircle $\Omega$ . Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$ . Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$ . Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$ . Prove that the points $D,G$ and $X$ are collinear.

Proposed by Ismail Isaev and Mikhail Isaev, Russia

Problem 1

by SpectralS, Jul 10, 2012, 5:24 PM

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$ , and to the lines $AB$ and $AC$ at $K$ and $L$ , respectively. The lines $LM$ and $BJ$ meet at $F$ , and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$ , and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$ , to the ray $AB$ beyond $B$ , and to the ray $AC$ beyond $C$ .)

Proposed by Evangelos Psychas, Greece

geometry

IMO Shortlist

Prove perpendicular

by shobber, Apr 1, 2006, 10:42 AM

Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$ . Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$ , respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced.

Prove that $QO$ is perpendicular to $BC$ .

perpendicular bisector

Two circles, a tangent line and a parallel

by Valentin Vornicu, Oct 24, 2005, 10:15 AM

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$ . Let $AB$ be the line tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let $CD$ be the line parallel to $AB$ and passing through the point $M$ , with $C$ on $G_1$ and $D$ on $G_2$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Show that $EP = EQ$ .

Submit

I still exist as well.
by G.G.Otto, Aug 11, 2023, 2:44 AM
hello I'm still here lol
by player01, Aug 6, 2022, 6:24 PM
[REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...
by sonone, Feb 18, 2022, 10:29 PM
wow its been like 2.5 years since geo class
by pieMax2713, Feb 4, 2022, 8:38 PM
@violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment
by sonone, Aug 19, 2021, 12:45 AM
ORZ CODER
by samrocksnature, Aug 9, 2021, 9:57 PM
Could you make more Asymptote lessons on your "How to do Asymptote" blog?
by violin21, Aug 9, 2021, 7:26 PM
You can take it, just C&P the CSS into your CSS area
by sonone, Apr 17, 2021, 10:08 PM
how can we take the CSS if we have permission to not take it?
by GoogleNebula, Apr 17, 2021, 5:22 PM
That is awesome!
by sonone, Apr 15, 2021, 10:09 PM
I modified your dodecahedron and got:
[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
by Andrew2019, Mar 26, 2021, 12:15 AM
Not too many, just changing the color here and there. I really like your CSS!
by sonone, Feb 2, 2021, 10:35 AM
Nice!

I see you're making changes to the CSS.
by G.G.Otto, Feb 1, 2021, 9:26 PM
I'm learning Java now!
by sonone, Feb 1, 2021, 5:56 PM
And I took part of it from CaptainFlint and then added a ton of modifications.
by G.G.Otto, Dec 1, 2020, 8:56 AM
fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
by sonone, Apr 25, 2020, 8:42 PM
When it says "At least 11 users clicked that button" it's incorrect since one person could have just posted 11 times. It should say "At most 11 users click that button".
by G.G.Otto, Apr 25, 2020, 8:26 PM
see this PM
by sonone, Apr 25, 2020, 7:03 PM
I can't get the cursor to work.
by G.G.Otto, Apr 25, 2020, 6:38 PM
If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
by sonone, Apr 23, 2020, 5:30 PM
Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
by G.G.Otto, Apr 22, 2020, 11:15 PM
You should download it, as it's really useful when learning python. You can find out more here
by G.G.Otto, Apr 19, 2020, 10:04 PM
No. I don't have IDLE.
by sonone, Apr 18, 2020, 7:49 PM
Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
by G.G.Otto, Apr 14, 2020, 6:47 AM
I don't know how to change it....
by G.G.Otto, Apr 5, 2020, 5:58 PM
Do you know how the change the into thing above my avatar a different color? It's too hard to see.
by sonone, Apr 5, 2020, 2:08 PM
WELCOME TO MY BLOG!
by sonone, Mar 31, 2020, 4:56 PM

98 shouts

sonone's place

by Melid, Apr 23, 2025, 3:01 PM

by NO_SQUARES, Apr 23, 2025, 9:08 AM

by Dadgarnia, Mar 12, 2020, 10:54 AM

by DylanN, Apr 9, 2019, 6:54 AM

by ipaper, Jul 9, 2014, 11:38 AM

by v_Enhance, Jul 30, 2013, 5:19 AM

by WakeUp, Jul 13, 2012, 11:41 AM

by SpectralS, Jul 10, 2012, 5:24 PM

by shobber, Apr 1, 2006, 10:42 AM

by Valentin Vornicu, Oct 24, 2005, 10:15 AM