Finding the set of all triangles satisfying the given condition
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let $S$ be the set of all triangles $ABC$ for which
$ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.
After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation
$0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$
How do I solve this equation to get the required result?
geometry triangle
add a comment |Â
up vote
0
down vote
favorite
Let $S$ be the set of all triangles $ABC$ for which
$ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.
After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation
$0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$
How do I solve this equation to get the required result?
geometry triangle
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $S$ be the set of all triangles $ABC$ for which
$ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.
After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation
$0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$
How do I solve this equation to get the required result?
geometry triangle
Let $S$ be the set of all triangles $ABC$ for which
$ 5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,$
where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. How do I Prove that all triangles in $S$ are isosceles and similar to one another.
After taking $s-a=x$, $s-b=y$, $s-c=z$ and assuming WLWOG that $x le y le z$ after much of grueling work I got an equation
$0 = 4y^2z^2 + 25x^2y^2 + 25x^2z^2 - 16xy^2z - 16xyz^2 + 14x^2yz$
How do I solve this equation to get the required result?
geometry triangle
geometry triangle
edited Sep 10 at 13:26
asked Sep 7 at 9:03
saisanjeev
517212
517212
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
This is your answer
I didn't write the answer cause it would take much time , sorry for that.
Can you please provide the full solution as google books is not allowing me to view the complete solution
â saisanjeev
Sep 10 at 13:28
I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
â Subhajit Halder
Sep 10 at 13:32
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
This is your answer
I didn't write the answer cause it would take much time , sorry for that.
Can you please provide the full solution as google books is not allowing me to view the complete solution
â saisanjeev
Sep 10 at 13:28
I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
â Subhajit Halder
Sep 10 at 13:32
add a comment |Â
up vote
0
down vote
The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
This is your answer
I didn't write the answer cause it would take much time , sorry for that.
Can you please provide the full solution as google books is not allowing me to view the complete solution
â saisanjeev
Sep 10 at 13:28
I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
â Subhajit Halder
Sep 10 at 13:32
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
This is your answer
I didn't write the answer cause it would take much time , sorry for that.
The question you wrote is wrong, it should be $$5left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r,\ and not left( dfrac1AP + dfrac1BQ + dfrac1CR right) - dfrac3min AP, BQ, CR = dfrac6r, $$
This is your answer
I didn't write the answer cause it would take much time , sorry for that.
answered Sep 7 at 9:22
Subhajit Halder
1039
1039
Can you please provide the full solution as google books is not allowing me to view the complete solution
â saisanjeev
Sep 10 at 13:28
I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
â Subhajit Halder
Sep 10 at 13:32
add a comment |Â
Can you please provide the full solution as google books is not allowing me to view the complete solution
â saisanjeev
Sep 10 at 13:28
I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
â Subhajit Halder
Sep 10 at 13:32
Can you please provide the full solution as google books is not allowing me to view the complete solution
â saisanjeev
Sep 10 at 13:28
Can you please provide the full solution as google books is not allowing me to view the complete solution
â saisanjeev
Sep 10 at 13:28
I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
â Subhajit Halder
Sep 10 at 13:32
I just searched for it and posted the link , ok then please wait and let me try the solution myself . I am just a 12 th class student.
â Subhajit Halder
Sep 10 at 13:32
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2908419%2ffinding-the-set-of-all-triangles-satisfying-the-given-condition%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password