Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
geometry
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
Balkan MO 2022/1 is reborn
Assassino9931   7
N an hour ago by Rayvhs
Source: Bulgaria EGMO TST 2023 Day 1, Problem 1
Let $ABC$ be a triangle with circumcircle $k$. The tangents at $A$ and $C$ intersect at $T$. The circumcircle of triangle $ABT$ intersects the line $CT$ at $X$ and $Y$ is the midpoint of $CX$. Prove that the lines $AX$ and $BY$ intersect on $k$.
7 replies
Assassino9931
Feb 7, 2023
Rayvhs
an hour ago
J
No more topics!
H
J
H
tough question
Bhashwar   3
N Apr 30, 2015 by jayme
Source: group mathematical olympiad 2014 india
Let ABC be an acute angled triangle & let I be its incentre. Let the incircle of triangle ABC touch BC in D. The incircle of the triangle ABD touches AB in E ; the incircle of the triangle ACD touches BC in F. Prove that B, E, I, F are concyclic.
3 replies
Bhashwar
Apr 30, 2015
jayme
Apr 30, 2015
J
tough question
G H J
Reveal topic tags
geometry
L
G H BBookmark kLocked kLocked NReply
Source: group mathematical olympiad 2014 india
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Bhashwar
1 post
Bhashwar
#1 h  Apr 30, 2015, 6:02 AM • 1 Y
Y by Adventure10
Let ABC be an acute angled triangle & let I be its incentre. Let the incircle of triangle ABC touch BC in D. The incircle of the triangle ABD touches AB in E ; the incircle of the triangle ACD touches BC in F. Prove that B, E, I, F are concyclic.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sunken rock
4394 posts
sunken rock
#3 h  Apr 30, 2015, 8:39 AM • 2 Y
Y by Adventure10, Mango247
hint only

Best regards,
sunken rock
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
andria
824 posts
andria
#4 h  Apr 30, 2015, 9:35 AM • 2 Y
Y by Adventure10, Mango247
My solution: let the incircle touch $AB,AC$ at $T,S$ note that $TE=AE-AT=\frac{AB+AD-BD}{2}-\frac{AB+AC-BC}{2}=\frac{AD-AC+(BC-BD)}{2}=\frac{AD+CD-AC}{2}=DF$ so $TE=DF$ because $IT=ID=r$ we get that $\triangle ITE=\triangle IDE \longrightarrow \angle IFB=\angle ITB$ so $BEIF$ is cyclic. DONE
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jayme
9792 posts
jayme
#5 h  Apr 30, 2015, 11:59 AM • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,
some ideas...
1. the two incircle are tangnet to AD at the same point
2. the points of contact F and F'of the two incircles are symmetric wrt D
3. consider the circle going through F, I and B
4. by considering trapeze, we can prove that E, F' and Z (intersection of ID and the last circle) are collinear
5 by considering a parallelogram and a trapeze, E is on the last circle...

Sincerely
Jean-Louis
Z K Y
N Quick Reply
G
H
=
a