A tangent problem

by hn111009, May 10, 2025, 2:09 AM

Let quadrilateral $ABCD$ with $P$ be the intersection of $AC$ and $BD.$ Let $\odot(APD)$ meet again $\odot(BPC)$ at $Q.$ Called $M$ be the midpoint of $BD.$ Assume that $\angle{DPQ}=\angle{CPM}.$ Prove that $AB$ is the tangent of $\odot(APD)$ and $BC$ is the tangent of $\odot(AQB).$

geometry

Lines concur on bisector of BAC

by Invertibility, May 9, 2025, 8:19 PM

Let $\Omega$ be the circumcircle of a scalene triangle $ABC$ . Let $\omega$ be a circle internally tangent to $\Omega$ in $A$ . Tangents from $B$ touch $\omega$ in $P$ and $Q$ , such that $P$ lies in the interior of $\triangle{}ABC$ . Similarly, tangents from $C$ touch $\omega$ in $R$ and $S$ , such that $R$ lies in the interior of $\triangle{}ABC$ .

Prove that $PS$ and $QR$ concur on the bisector of $\angle{}BAC$ .

geometry

Interesting functional equation with geometry

by User21837561, May 9, 2025, 8:14 AM

For an acute triangle $ABC$ , let $O$ be the circumcentre, $H$ be the orthocentre, and $G$ be the centroid.
Let $f:\pi\rightarrow\mathbb R$ satisfy the following condition:
$f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$
Prove that $f$ is constant.

This post has been edited 2 times. Last edited by User21837561, Yesterday at 4:10 PM
Reason: Wrong year for the source

algebra

functional equation

geometry

Problem 4 of Finals

by GeorgeRP, Sep 10, 2024, 4:39 PM

The diagonals $AD$ , $BE$ , and $CF$ of a hexagon $ABCDEF$ inscribed in a circle $k$ intersect at a point $P$ , and the acute angle between any two of them is $60^\circ$ . Let $r_{AB}$ be the radius of the circle tangent to segments $PA$ and $PB$ and internally tangent to $k$ ; the radii $r_{BC}$ , $r_{CD}$ , $r_{DE}$ , $r_{EF}$ , and $r_{FA}$ are defined similarly. Prove that
$r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}.$

geometry

junior perpenicularity, 2 circles related

by parmenides51, Mar 2, 2024, 7:55 PM

Consider an acute triangle $ABC$ and it's circumcircle $\omega$ . With center $A$ , we construct a circle $\gamma$ that intersects arc $AB$ of circle $\omega$ , that doesn't contain $C$ , at point $D$ and arc $AC$ , that doesn't contain $B$ , at point $E$ . Suppose that the intersection point $K$ of lines $BE$ and $CD$ lies on circle $\gamma$ . Prove that line $AK$ is perpendicular on line $BC$ .

This post has been edited 1 time. Last edited by parmenides51, Mar 2, 2024, 7:55 PM

geometry

perpendicular

An I for an I

by Eyed, Jul 20, 2021, 8:54 PM

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$ . Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$ . Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$ , and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$ . Rays $PM$ and $QN$ meet at $X$ . Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$ .

Show that $A,X,Y$ are collinear.

This post has been edited 1 time. Last edited by Eyed, Jul 20, 2021, 9:59 PM

I accidentally drew a 200-gon and ran out of time

by justin1228, Oct 25, 2020, 10:59 PM

The sides of a convex $200$ -gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion.
Suppose the extensions of the red sides determine a regular $100$ -gon, as do the extensions of the blue sides.

Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}},\ \dots, \ \overline{A_{99}A_{199}}$ are concurrent.

Proposed by: Ankan Bhattacharya

This post has been edited 2 times. Last edited by justin1228, Oct 25, 2020, 11:10 PM
Reason: ankan

geometry

USEMO

USEMO 2020

GHM and ABC are tangent to each other.

by IndoMathXdZ, Jul 12, 2020, 6:11 PM

Given a triangle $ABC$ with circumcenter $O$ and orthocenter $H$ . Line $OH$ meets $AB, AC$ at $E,F$ respectively.
Define $S$ as the circumcenter of $AEF$ . The circumcircle of $AEF$ meets the circumcircle of $ABC$ again at $J$ , $J \not= A$ . Line $OH$ meets circumcircle of $JSO$ again at $D$ , $D \not= O$ and circumcircle of $JSO$ meets circumcircle of $ABC$ again at $K$ , $K \not= J$ . Define $M$ as the intersection of $JK$ and $OH$ and $DK$ meets circumcircle of $ABC$ at points $K,G$ .

Prove that circumcircle of $GHM$ and circumcircle of $ABC$ are tangent to each other.

Proposed by 郝敏言, China

angle chasing with 2 midpoints, equal angles given and wanted

by parmenides51, Dec 11, 2018, 10:02 PM

In the triangle $ABC$ , ${{A}_{1}}$ and ${{C}_{1}}$ are the midpoints of sides $BC$ and $AB$ respectively. Point $P$ lies inside the triangle. Let $\angle BP {{C}_{1}} = \angle PCA$ . Prove that $\angle BP {{A}_{1}} = \angle PAC$ .

This post has been edited 1 time. Last edited by parmenides51, Mar 26, 2024, 2:37 PM

IMO 2018 Problem 1

by juckter, Jul 9, 2018, 11:20 AM

Let $\Gamma$ be the circumcircle of acute triangle $ABC$ . Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$ . The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece

This post has been edited 2 times. Last edited by djmathman, Jun 16, 2020, 4:02 AM
Reason: problem author

Keith50

by hn111009, May 10, 2025, 2:09 AM

by Invertibility, May 9, 2025, 8:19 PM

by User21837561, May 9, 2025, 8:14 AM

by GeorgeRP, Sep 10, 2024, 4:39 PM

by parmenides51, Mar 2, 2024, 7:55 PM

by Eyed, Jul 20, 2021, 8:54 PM

by justin1228, Oct 25, 2020, 10:59 PM

by IndoMathXdZ, Jul 12, 2020, 6:11 PM

by parmenides51, Dec 11, 2018, 10:02 PM

by juckter, Jul 9, 2018, 11:20 AM