Hardest geometry question in history answered by student trivially............ How? (1 Viewer)

no_arg

Member
Joined
Mar 25, 2004
Messages
67
Gender
Undisclosed
HSC
N/A
Suppose that PQR is an equilateral triangle with an interior point X. Let angle PXR = s and angle QXR = t. Find (in terms of s and t) the three angles of any triangle with side lengths equal to PX, QX and RX.
 
Last edited:

Drdusk

Moderator
Moderator
Joined
Feb 24, 2017
Messages
2,025
Location
a VM
Gender
Male
HSC
2018
Uni Grad
2023
I think it's because X must only be in the center for a triangle to form?
 

no_arg

Member
Joined
Mar 25, 2004
Messages
67
Gender
Undisclosed
HSC
N/A
I think it's because X must only be in the center for a triangle to form?
No X can be any interior point. PX QX and RX need to be lifted out of the diagram to form another triangle.
 

fan96

617 pages
Joined
May 25, 2017
Messages
543
Location
NSW
Gender
Male
HSC
2018
Uni Grad
2024
No X can be any interior point. PX QX and RX need to be lifted out of the diagram to form another triangle.
Must the angles be expressed only in terms of and ?

The best I could do is





where is the side length of the equilateral triangle and are the angles of the newly formed triangle.

(Hopefully that's correct...)
 

no_arg

Member
Joined
Mar 25, 2004
Messages
67
Gender
Undisclosed
HSC
N/A
Must the angles be expressed only in terms of and ?

The best I could do is





where is the side length of the equilateral triangle and are the angles of the newly formed triangle.

(Hopefully that's correct...)
Given similarity, the solution cannot depend on the side length of the original triangle. Answer is disturbingly simple and the proof is stunning.
 

quickoats

Well-Known Member
Joined
Oct 26, 2017
Messages
969
Gender
Undisclosed
HSC
2019
 

no_arg

Member
Joined
Mar 25, 2004
Messages
67
Gender
Undisclosed
HSC
N/A
Simply rotate the entire diagram about P anticlockwise by sixty degrees and voila!
 

fan96

617 pages
Joined
May 25, 2017
Messages
543
Location
NSW
Gender
Male
HSC
2018
Uni Grad
2024
Given similarity, the solution cannot depend on the side length of the original triangle. Answer is disturbingly simple and the proof is stunning.
The quantity



can probably be simplified so as to remove .

The answer I gave holds numerically for all such equilateral and isosceles triangles formed and most likely for the rest of them too.

The other solutions are definitely much nicer though.

Both proofs a little clumsy
I think the first answer given is essentially the same as the one you posted, just a bit more direct.
 

no_arg

Member
Joined
Mar 25, 2004
Messages
67
Gender
Undisclosed
HSC
N/A
The quantity



can probably be simplified so as to remove .

The answer I gave holds numerically for all such equilateral and isosceles triangles formed and most likely for the rest of them too.

The other solutions are definitely much nicer though.


I think the first answer given is essentially the same as the one you posted, just a bit more direct.
The use of congruence makes it less direct.
 

jyu

Member
Joined
Nov 14, 2005
Messages
623
Gender
Male
HSC
2006
(s-60), (t-60), (300-s-t)
 
Last edited:

jyu

Member
Joined
Nov 14, 2005
Messages
623
Gender
Male
HSC
2006
Suppose that PQR is a 50-60-70 (respectively) triangle with an interior point X. Let angle PXR = s and angle QXR = t.
Find (in terms of s and t) the three angles of any triangle with side lengths equal to PX, QX and RX.
 
Last edited:

idkkdi

Well-Known Member
Joined
Aug 2, 2019
Messages
2,454
Gender
Male
HSC
2021
Suppose that PQR is a 50-60-70 (respectively) triangle with an interior point X. Let angle PXR = s and angle QXR = t.
Find (in terms of s and t) the three angles of any triangle with side lengths equal to PX, QX and RX.
so is anyone going to solve this?
 

idkkdi

Well-Known Member
Joined
Aug 2, 2019
Messages
2,454
Gender
Male
HSC
2021
so is anyone going to solve this?
Suppose that PQR is a 50-60-70 (respectively) triangle with an interior point X. Let angle PXR = s and angle QXR = t.
Find (in terms of s and t) the three angles of any triangle with side lengths equal to PX, QX and RX.
Can someone please solve this lol.
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top