geometry problems from Discord Geometry Olympiad (DGO), with aops links in the names
collected inside aops here
2021
Let $ABC$ be a triangle and let $M$ be a point on side $BC$ such that $BM < MC$. Let $B', C'$ be points on sides $AC, AB$ such that $AM, BB', CC'$ are concurrent. Determine if it is possible for $B'C'$ to be parallel to $BC$.
by wassupevery1
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. $OH$ and $OA$ intersect circumcircle of triangle $BOC$ again at $P, K$. Suppose that $P, O$ lie on the same side with respect to $BC$. Prove that $P B + PC = PK$.
by wassupevery1
Let triangle $ABC$ be a triangle with incenter $I$ and circumcircle $\Omega$ with circumcenter $O$. The incircle touches $CA, AB$ at $E, F$ respectively. $R$ is another intersection point of external bisector of $\angle BAC$ with $\Omega$, and $T$ is $\text{A-mixtillinear}$ incircle touch point to $\Omega$. Let $W, X, Z$ be points lie on $\Omega$. $RX$ intersect $AI$ at $Y$ . Assume that $R \ne X$. Suppose that $E, F, X, Y$ and $W, Z, E, F$ are concyclic, and $AZ, EF, RX$ are concurrent.
Prove that
$\bullet$ $AZ, RW, OI$ are concurrent.
$\bullet$ $\text{A-symmedian}$, tangent line to $\Omega$ at $T$ and $WZ$ are concurrent.
by wassupevery1 and k12byda5h
Let $M$ be an interior point inside a triangle $ABC$ such that $\angle MAB < \angle MAC$. Is it always true that
$\frac{MB}{MC}<\frac{AB}{AC}$ ?
by wassupevery1
For each triangle $XYZ$, we say that a triangle $ABC$ is relatively good triangle of triangle $XYZ$ if and only if $A, B, C$ lie on side $YZ, XZ, XY$ respectively, and $XA, YB, ZC$ are concurrent. Given a triangle $ABC$, let $A_1B_1C_1$ to be a relatively good triangle of triangle $ABC$. Define triangle $A_nB_nC_n$ to be relatively good of triangle $A_{n-1}B_{n-1}C_{n-1}$ for every positve integer $n$ greater than $1$. Determine all integers $n\geq 2$ such that $AA_n, BB_n, CC_n$ are always concurrent.
by wassupevery1
Let $O$ be the circumcenter of triangle $ABC$. The altitudes from $A, B, C$ of triangle $ABC$ intersects the circumcircle of the triangle $ABC$ at $A_1, B_1, C_1$ respectively. $AO, BO, CO$ meets $BC, CA, AB$ at $A_2, B_2, C_2$ respectively. Prove that the circumcircles of triangles $AA_1A_2, BB_1B_2, CC_1C_2$ share two common points.
by wassupevery1
Given triangle $ABC$ with orthocenter $H$ and $AD,BE,CF$ be the altitudes. Let $K$ be the symmedian point of triangle $ABC$. $N$ is the circumcenter of triangle $DEF$. $L$ is the isogonal conjugate point of $K$ with respect to triangle $DEF$. Prove that $H,N,L$ are collinear.
Let triangle $ABC$ be the triangle with circumcenter $O$. $P$ be the point on circumcircle of triangle $BOC$. $AP,AO$ intersect the circle $BOC$ again at $X,Y$. Suppose that $PY,OX,BC$ are concurrent. Suppose that $P,O$ lie on the same side with respect to $BC$. Prove that $PB+PC = PY$.
by wassupevery1 and k12byda5h
Let triangle $ABC$ be a triangle with incenter $I$ and circumcircle $\Omega$ with circumcenter $O$. The incircle touches $CA, AB$ at $E, F$ respectively. $R$ is another intersection point of external bisector of $\angle BAC$ with $\Omega$, and $T$ is $\text{A-mixtillinear}$ incircle touch point to $\Omega$. Let $W, X, Z$ be points lie on $\Omega$. $RX$ intersect $AI$ at $Y$ . Assume that $R \ne X$. Suppose that $E, F, X, Y$ and $W, Z, E, F$ are concyclic, and $AZ, EF, RX$ are concurrent.
Prove that
$\bullet$ $AZ, RW, OI$ are concurrent.
$\bullet$ $\text{A-symmedian}$, tangent line to $\Omega$ at $T$ and $WZ$ are concurrent.
by wassupevery1 and k12byda5h
Let triangle $ABC$ be triangle with orthocenter $H$ and circumcircle $O$. A point $X$ lies on line $BC$. $AH$ intersects the circumcircle of triangle $ABC$ again at $H'$. $AX$ intersects circumcircle of triangle $H'HX$ again at $Y$ and intersects circumcircle of triangle $ABC$ again at $Z$. Let $G$ be the intersection of $BC$ with $H'O$. Let $P$ lies on $AB$ such that $PH'A = 90^\circ - \angle BAC$. Prove that
1. the ratio and the angle between $YH$ and $ZG$ do not depend on the choices of $X$.
2. $\angle PYH = \angle BZG$.
by k12byda5h
Given a right triangle $ABC$ ($\angle CAB = 90^\circ$) with $AC > AB$. Points $X,Y$ lie on the line $AB$ such that $X,A,B,Y$ lie in this order. Point $Z,W$ lie on the line $AC$ such that $A,C,Z,W$ lie in this order. Suppose that $BX = BY = AC,CZ=AX,CW = AY$. Prove that the quadrilateral formed by the lines $CX,CY,BW,BZ$ is cyclic.
Let $O$ be the circumcenter of triangle $ABC$. The altitudes from $A, B, C$ of triangle $ABC$ intersects the circumcircle of the triangle $ABC$ at $A_1, B_1, C_1$ respectively. $AO, BO, CO$ meets $BC, CA, AB$ at $A_2, B_2, C_2$ respectively. Prove that the circumcircles of triangles $AA_1A_2, BB_1B_2, CC_1C_2$ share two common points.
by wassupevery1
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