Hagge circle, Thomson cubic, coaxal

by kosmonauten3114, May 29, 2025, 2:24 PM

Let $\triangle{ABC}$ be a scalene triangle, $\triangle{M_AM_BM_C}$ its medial triangle, and $P$ a point on the Thomson cubic (= $\text{K002}$ ) of $\triangle{ABC}$ . (Suppose that $P \notin \odot(ABC)$ ).
Let $\triangle{A'B'C'}$ be the circumcevian triangle of $P$ wrt $\triangle{ABC}$ .
Let $\triangle{P_AP_BP_C}$ be the pedal triangle of $P$ wrt $\triangle{ABC}$ .
Let $A_1$ be the reflection in $BC$ of $A'$ . Define $B_1$ , $C_1$ cyclically.
Let $A_2$ be the reflection in $M_A$ of $A'$ . Define $B_2$ , $C_2$ cyclically.
Let $A_3$ be the reflection in $P_A$ of $A'$ . Define $B_3$ , $C_3$ cyclically.

Prove that $\odot(A_1B_1C_1)$ , $\odot(A_2B_2C_2)$ , $\odot(A_3B_3C_3)$ and the orthocentroidal circle of $\triangle{ABC}$ are coaxal.

Attachments:

This post has been edited 1 time. Last edited by kosmonauten3114, 4 hours ago

geometry

Triangle

Hagge circle

Thomson cubic

orthocentroidal circle

coaxal circles

coaxal

strange geometry problem

by Zavyk09, May 28, 2025, 4:32 PM

Let $ABC$ be a triangle with circumcenter $O$ and internal bisector $AD$ . Let $AD$ cuts $(O)$ again at $M$ and $MO$ cuts $(O)$ again at $N$ . Point $L$ lie on $AD$ such that $(AD, LM) = -1$ . The line pass through $L$ and perpendicular to $AD$ intersects $NC, NB$ at $P, Q$ respectively. Let circumcircle of $\triangle NPQ$ cuts $(O)$ at $G \ne N$ . Prove that $\angle AGD = 90^{\circ}$ .

geometry

angle bisector

circumcircle

Geometry problem

by Whatisthepurposeoflife, May 28, 2025, 1:45 PM

Triangle ∆ABC is scalene the circle w that goes through the points A and B intersects AC at E BC at D let the Lines BE and AD intersect at point F. And let the tangents A and B of circle w Intersect at point G.
Prove that C F and G are collinear

This post has been edited 1 time. Last edited by Whatisthepurposeoflife, Yesterday at 1:48 PM
Reason: Needed to show the source of the problem but proving the original problem wouldn't prove this one.

geometry

Random concyclicity in a square config

by Maths_VC, May 27, 2025, 7:38 PM

Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$ , and let $N$ be the intersection point of segments $AC$ and $DM$ . The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$ , $C$ , $P$ and $Q$ lie on a single circle.

geometry

circumcircle

Nine point circle + Perpendicularities

by YaoAOPS, Mar 5, 2025, 3:28 AM

Suppose $\triangle ABC$ has $D$ as the midpoint of $BC$ and orthocenter $H$ . Let $P$ be an arbitrary point on the nine point circle of $ABC$ . The line through $P$ perpendicular to $AP$ intersects $BC$ at $Q$ . The line through $A$ perpendicular to $AQ$ intersects $PQ$ at $X$ . If $M$ is the midpoint of $AQ$ , show that $HX \perp DM$ .

geometry

TST

Iran 3rd Round Geo

by M11100111001Y1R, Aug 23, 2024, 11:09 AM

Consider an acute scalene triangle $\triangle{ABC}$ . The interior bisector of $A$ intersects $BC$ at $E$ and the minor arc of $\overarc {BC}$ in circumcircle of $\triangle{ABC}$ at $M$ . Suppose that $D$ is a point on the minor arc of $\overarc {BC}$ such that $ED=EM$ . $P$ is a point on the line segment of $AD$ such that $\angle ABP=\angle ACP \not= 0$ . $O$ is the circumcenter of $\triangle{ABC}$ . Prove that $OP \perp AM$ .

This post has been edited 2 times. Last edited by M11100111001Y1R, Aug 29, 2024, 11:04 AM

geometry

circumcircle

Geometry One Problem Bounty Hunt Contest: Trapezium Geometry

by anantmudgal09, Jun 3, 2024, 11:59 AM

Let $ABC$ be an acute-angled scalene triangle with incircle $\omega$ and circumcircle $\Gamma$ . Suppose $\omega$ touches line $BC$ at $D$ and the tangent to $\Gamma$ at $A$ meets line $BC$ at $T$ . Two circles passing through $A$ and $T$ tangent to $\omega$ meet line $AD$ again at $X$ and $Y$ .

Prove that $BXCY$ is a trapezium.

geometry

trapezium

trapezoid

Tangential quadrilateral and 8 lengths

by popcorn1, Jul 20, 2021, 8:44 PM

Let $\Gamma$ be a circle with centre $I$ , and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$ . Let $\Omega$ be the circumcircle of the triangle $A I C$ . The extension of $B A$ beyond $A$ meets $\Omega$ at $X$ , and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$ . The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$ , respectively. Prove that $A D+D T+T X+X A=C D+D Y+Y Z+Z C.$
Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland

This post has been edited 1 time. Last edited by popcorn1, Jul 20, 2021, 9:56 PM

geometry

Find all possible values of BT/BM

by va2010, Jul 7, 2016, 8:59 PM

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$ . A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$ . Determine all possible values of $\frac{BT}{BM}$ .

This post has been edited 1 time. Last edited by v_Enhance, Jul 7, 2016, 9:23 PM
Reason: improve title

geometry

Midpoints of arcs form a similar triangle

by v_Enhance, Aug 13, 2013, 6:05 AM

Let $ABC$ be a triangle and $D$ , $E$ , $F$ be the midpoints of arcs $BC$ , $CA$ , $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$ . Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$ . Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$ . Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.

geometry

vector

parallelogram

geometric transformation

homothety

Submit

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