XYZW cyclic wanted, # BMGW , # CMFY, circle through B,C of triangle ABC

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

XYZW cyclic wanted, # BMGW , # CMFY, circle through B,C of triangle ABC

G H J

Reveal topic tags

cyclic quadrilateral

L

G H BBookmark kLocked kLocked NReply

Source: 2017 Iberoamerican Shortlist G6 https://artofproblemsolving.com/community/c2533295_iberoamerican_shortlist_2011_geometry

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30629 posts

#1 h Nov 5, 2021, 8:59 PM • 1 Y

Y by HWenslawski

Let $ABC$ be a triangle; a circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. Let $P$ be the point of intersection of $BE$ and $CD$ , the inner angle bisector of $\angle EPC$ intersects $AB$ and $AC$ at $F$ and $G$ respectively. Let $M$ be the midpoint of $FG$ , the parallelograms $BMGW$ and $CMFY$ are constructed. If $Z$ is the intersection of $WG$ and $FY$ and $AP$ intersects the circumcircle of triangle $ABC$ at $X \ne A$ , show that quadrilateral $XYZW$ it is cyclic.

This post has been edited 1 time. Last edited by parmenides51, Nov 5, 2021, 9:07 PM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

1079 posts

#2 h Nov 7, 2021, 2:13 AM • 1 Y

Y by parmenides51

The crux of this problem is finding out what point $X$ actually is. I contend that $X$ is the Miquel point of quadrilateral $AFZG$ .
Thus, I'll solve the following equivalent problem instead:

Restated problem wrote:

Consider quadrilateral $AFZG$ with $AF=AG$ . Let $M$ be the midpoint of $\overline{FG}$ . Let $T=\overline{AF}\cap\overline{ZG}$ and $U=\overline{AG}\cap\overline{ZF}$ . Let $X$ be the Miquel point of complete quadrilateral $AFTZUG$ . Let $P=\overline{AX}\cap\overline{FG}$ . Let $B,C,Y,W$ be the midpoints of $\overline{TF},\overline{UG},\overline{UF},\overline{TG}$ . Prove that $X$ lies on $(YZW)$ and $(ABC)$ and $\angle FPB=\angle CPG$ .

By spiral sim $X$ lies on $(YZW)$ and $(ABC)$ as $\triangle XTF\sim\triangle XGU$ and $\triangle XTG\sim\triangle XFU$ . Furthermore, $\begin{align*} \frac{PF}{PG}=\frac{\frac{AF\cdot\sin{\angle PAF}}{\sin{\angle APF}}}{\frac{AG\cdot\sin{\angle PAG}}{\sin{\angle APG}}}=\frac{\sin{\angle PAF}}{\sin{\angle PAG}}=\frac{\sin{\angle XAF}}{\sin{\angle XAU}}=\frac{XF}{XU}=\frac{TF}{GU}=\frac{BF}{GC}. \end{align*}$ Therefore, $\triangle FPB\sim\triangle GPC\implies \angle FPB=\angle CPG$ .

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

241 posts

#3 h Nov 7, 2021, 11:05 PM • 1 Y

Y by parmenides51

This problem was proposed by me

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

109 posts

#4 h Nov 8, 2021, 2:13 AM

Y by

Hello. Where can I find the full shortlist ?

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30629 posts

#5 h Nov 8, 2021, 10:54 AM

Y by

These shortlists are not made public, I have posted only the geometry ones I could find , a few older Ibero shortlists are made public in this site (currently unavailable)

This post has been edited 1 time. Last edited by parmenides51, Nov 8, 2021, 11:21 AM

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a