Probably a good lemma

by Zavyk09, May 18, 2025, 12:50 PM

Let $ABC$ be a triangle with circumcircle $\omega$ . Arbitrary points $E, F$ on $AC, AB$ respectively. Circumcircle $\Omega$ of triangle $AEF$ intersects $\omega$ at $P \ne A$ . $BE$ intersects $CF$ at $I$ . $PI$ cuts $\Omega$ and $\omega$ at $K, L$ respectively. Construct parallelogram $QFRE$ . Prove that $A, R, P$ are collinear.

geometry

circumcircle

parallelogram

collinear

Miquel point

geometry

by gggzul, May 18, 2025, 9:37 AM

Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$ . $D$ is the midpoint of $AC$ . Let the angle bisector of $\angle ACB$ cut $BD$ at $P$ and $G$ be the centroid of $ABC$ . $(CPG)$ meets $BC$ at $Q\ne C$ and $R$ is the projection of $Q$ onto $AB$ . Prove that $R, G, P, A$ lie on a common circle.

geometry

angle bisector

Gergonne point Harmonic quadrilateral

by niwobin, May 17, 2025, 8:17 PM

Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)

Attachments:

geometry

Incircle in an isoscoles triangle

by Sadigly, May 16, 2025, 9:21 PM

Let $ABC$ be an isosceles triangle with $AB=AC$ , and let $I$ be its incenter. Incircle touches sides $BC,CA,AB$ at $D,E,F$ , respectively. Foot of altitudes from $E,F$ to $BC$ are $X,Y$ , respectively. Rays $XI,YI$ intersect $(ABC)$ at $P,Q$ , respectively. Prove that $(PQD)$ touches incircle at $D$ .

geometry

incenter

Prove that the triangle is isosceles.

by TUAN2k8, May 16, 2025, 9:55 AM

Given acute triangle $ABC$ with two altitudes $CF$ and $BE$ .Let $D$ be the point on the line $CF$ such that $DB \perp BC$ .The lines $AD$ and $EF$ intersect at point $X$ , and $Y$ is the point on segment $BX$ such that $CY \perp BY$ .Suppose that $CF$ bisects $BE$ .Prove that triangle $ACY$ is isosceles.

geometry

Triangle

A very beautiful geo problem

by TheMathBob, Mar 29, 2023, 2:16 PM

Given an acute triangle $ABC$ with their incenter $I$ . Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$ . Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$ . It is given that $\angle AIX = \angle XYA = 120^\circ.$ Prove that $YI$ is the angle bisector of $XYA$ .

This post has been edited 1 time. Last edited by TheMathBob, Mar 29, 2023, 2:24 PM

geometry

incenter

angle bisector

Circle is tangent to circumcircle and incircle

by ABCDE, Jun 24, 2016, 2:05 PM

Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$ , let its incircle be tangent to sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$ , respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$ . Finally, let $\gamma$ be the circumcircle of $\triangle AST$ .

(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$ .

(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$ .

James Lin

This post has been edited 1 time. Last edited by ABCDE, Jun 24, 2016, 2:07 PM

incircle excenter midpoints

by danepale, Sep 21, 2014, 2:15 PM

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$ . Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$ . Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.

Prove that the points $B, C, N,$ and $L$ are concyclic.

geometry

geometric transformation

reflection

homothety

perpendicular bisector

geometry proposed

Locus of Mobile points on Circle and Square

by Kunihiko_Chikaya, Feb 28, 2012, 2:58 AM

In the $xyz$ -plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$ -plane.

(1) Let $P(0,\ 0,\ 2)$ . When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$ ,
find the equation of the locus of the point $R$ .

(2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$ . When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$ , draw the domain swept by the point $R$ on the $xy$ -plane, then find the area.

Symmedian line

by April, May 10, 2009, 3:49 PM

Let be given a triangle $ABC$ and its internal angle bisector $BD$ $(D\in BC)$ . The line $BD$ intersects the circumcircle $\Omega$ of triangle $ABC$ at $B$ and $E$ . Circle $\omega$ with diameter $DE$ cuts $\Omega$ again at $F$ . Prove that $BF$ is the symmedian line of triangle $ABC$ .

geometric transformation

Submit

hi guys $~~~$
by Yiyj1, Apr 9, 2025, 7:25 AM
You will be remembered
by giangtruong13, Feb 26, 2025, 3:38 PM
2025 shout!
by just_a_math_girl, Jan 12, 2025, 7:25 AM
2024 shout
by bachkieu, Aug 22, 2024, 12:52 AM
helooooooooooo
by owenccc, Sep 27, 2023, 12:59 AM
hullo :<
by gracemoon124, Jul 14, 2023, 2:33 AM
hello $\text{[math]}$
by LeoLionTank, Feb 17, 2023, 9:02 PM
Halo thear
by HoRI_DA_GRe8, Oct 18, 2022, 7:44 AM
hi!!
just found this and I can't wait to read more!
so happy to have found this blog!
by Morrigan_Black, Jan 28, 2022, 1:05 PM
still waiting for the mathlinks camp lol
by CinarArslan, Jan 9, 2022, 1:55 PM
hello
by CyclicISLscelesTrapezoid, Nov 29, 2021, 6:36 PM
Right below the shout box it says how many it has.
by pith0n, May 11, 2021, 5:08 AM
Oh really the blog has 100 posts! I never counted the number of posts here. If I get some free time I will create a new page on my wordpress website and there I will post all the contents of this blog. So, make sure that you check the wordpress site.
by adityaguharoy, Mar 21, 2021, 2:58 PM
Nice blog!
by DCode10, Mar 10, 2021, 7:00 PM
Hi adityaguharoy! Nice blog!
by masadca, Feb 4, 2021, 9:19 PM
Nice blog!
by Mathisfun04, Dec 2, 2016, 12:00 AM
Nice blog!!!
by Ilikeapos, Aug 24, 2016, 2:56 PM
really nice CSS
by StarFrost7, Aug 16, 2016, 3:37 AM
hi!
Post some more!
Also nice css
by Intellectuality123, Aug 12, 2016, 11:35 PM
Hello!!!!!
by MinionLA, Jul 18, 2016, 9:15 PM
hi everyone
by matthMaster, Jul 2, 2016, 1:36 AM
You have a great blog here; just add some more stuff and it will be even better!
by bobert1, Jun 30, 2016, 9:29 PM
yeah sure we must and will discuss real numbers.
by adityaguharoy, Jun 20, 2016, 4:11 AM
Why not real numbers
by skipiano, Jun 17, 2016, 2:35 PM
very interesting topic !! ?? !!
by adityaguharoy, Jun 12, 2016, 6:52 AM
Hello!!!!!!! Very interesting topic, here!
by MinionLA, Jun 9, 2016, 11:43 PM
Should it not start by the discussion of Set Theory.
by alexanderoldspartan, Jun 9, 2016, 11:03 AM
Darn I missed it
by skipiano, Jun 9, 2016, 4:52 AM
first shout >
by doitsudoitsu, Apr 26, 2016, 8:03 PM

118 shouts

An Introduction to the Theory of Mathematics

by Zavyk09, May 18, 2025, 12:50 PM

by gggzul, May 18, 2025, 9:37 AM

by niwobin, May 17, 2025, 8:17 PM

by Sadigly, May 16, 2025, 9:21 PM

by TUAN2k8, May 16, 2025, 9:55 AM

by TheMathBob, Mar 29, 2023, 2:16 PM

by ABCDE, Jun 24, 2016, 2:05 PM

by danepale, Sep 21, 2014, 2:15 PM

by Kunihiko_Chikaya, Feb 28, 2012, 2:58 AM

by April, May 10, 2009, 3:49 PM