6 congruent triangles into convex hexagon - Ecuador TST 2022.1.3

Art of Problem Solving

AoPS Online

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

6 congruent triangles into convex hexagon - Ecuador TST 2022.1.3

G H J

Reveal topic tags

congruent triangles

L

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

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

30632 posts

#1 h Oct 25, 2022, 5:57 PM

Y by

The convex hexagon $ABCDEF$ satisfies that the triangles $ABC$ , $BCD$ , $CDE$ , $DEF$ , $EFA$ and $FAB$ are congruent. Show that $AD = BE = CF$ .

This post has been edited 1 time. Last edited by parmenides51, Oct 25, 2022, 5:58 PM

Z K Y

The post below has been deleted. Click to close.

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

1312 posts

#2 h Oct 26, 2022, 12:00 PM

Y by

If all inner angles of hexagon $ABCDEF$ are obtuse, then in $\triangle ABC,BCD$ , there must be $\angle ABC=\angle BCD$ . Consider the other pairs of congruent triangles similarly, we can get that all inner angles are equal. Moreover, they all equal to $120^\circ$ .
No matter $\triangle ABC\cong\triangle BCD$ or $\triangle ABC\cong\triangle BCD$ , we have $AB=CD$ . Consider the other pairs of congruent triangles similarly, we can get $AB=CD=EF,BC=DE=FA$ .
Construct the graph as attachment 1. Then the triangles $AGB,BHC,CID,DJE,ELF,FKA,GLI,KJH$ are equilateral, and some of them have the same side length as labeled. From
$\frac{HA}{AK}=\frac{HD}{DJ}=\frac{m+n}n,$ $AD\parallel KJ$ . Similarly, $BE\parallel HJ,CF\parallel KH$ . Now $\frac{AD}{KJ}=\frac{BE}{HJ}=\frac{CF}{KH}=\frac{n+m}{n+2m}.$ So $AD=BE=CF$ .
If there're any acute inner angles, say $\angle ABC$ , then triangles $ABC,FAB$ leads to $\angle ABC=\angle BAF$ , false.

Z K Y

The post below has been deleted. Click to close.

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

1312 posts

#3 h Oct 26, 2022, 12:02 PM

Y by

$~~~~~~~~~$

Attachments:

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a