For $H$ the orthocenter of $triangle ABC$, prove $|AH|h_a+|BH|h_b+|CH|h_c=frac12left(a^2+b^2+c^2right)$...
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Given acute $triangle ABC$ with orthocenter $H$, prove $$|AH|,h_a+|BH|,h_b+|CH|,h_c=frac12left(a^2+b^2+c^2right)$$
I tried to do something with the perimeter, area, half-perimeter, but I can't solve it. I am not good at contest problems, and I just started practicing. I'll be glad if I get an answer or just guidelines. I couldn't seem to find this problem solved or guidelines for it anywhere.
geometry
edited Nov 29 '18 at 21:17
Blue
47.8k870152
asked Nov 29 '18 at 20:54
PeroPero
1207
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closed as off-topic by T. Bongers, Théophile, KReiser, Alexander Gruber♦ Nov 30 '18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – T. Bongers, Théophile, KReiser, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Given acute $triangle ABC$ with orthocenter $H$, prove $$|AH|,h_a+|BH|,h_b+|CH|,h_c=frac12left(a^2+b^2+c^2right)$$
I tried to do something with the perimeter, area, half-perimeter, but I can't solve it. I am not good at contest problems, and I just started practicing. I'll be glad if I get an answer or just guidelines. I couldn't seem to find this problem solved or guidelines for it anywhere.
geometry
edited Nov 29 '18 at 21:17
Blue
47.8k870152
asked Nov 29 '18 at 20:54
PeroPero
1207
$endgroup$
closed as off-topic by T. Bongers, Théophile, KReiser, Alexander Gruber♦ Nov 30 '18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – T. Bongers, Théophile, KReiser, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Welcome to MSE. What work have you done so far?
$endgroup$
– Théophile
Nov 29 '18 at 21:00
$begingroup$
Please define all objects that appear, starting with the triangle and $ha$ (which may be some $h_a$, typed ash_ain LaTeX), in the text, not (only) in the title. Then please show the own efforts to solve. Else there are some people that downvote from the start.
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– dan_fulea
Nov 29 '18 at 21:01
$begingroup$
If you don't want your question to get downvoted, you should include more context such as your own attempt at a solution. math.meta.stackexchange.com/questions/26017/…
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– Ryan Greyling
Nov 29 '18 at 21:02
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What are $h_a$, $h_b$, $h_c$? Are they the lengths of the altitudes from $A$, $B$, $C$?
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– Blue
Nov 29 '18 at 21:18
$begingroup$
I believe so. It wasn't specified in the problem.
$endgroup$
– Pero
Nov 29 '18 at 21:20
add a comment |
$begingroup$
Given acute $triangle ABC$ with orthocenter $H$, prove $$|AH|,h_a+|BH|,h_b+|CH|,h_c=frac12left(a^2+b^2+c^2right)$$
I tried to do something with the perimeter, area, half-perimeter, but I can't solve it. I am not good at contest problems, and I just started practicing. I'll be glad if I get an answer or just guidelines. I couldn't seem to find this problem solved or guidelines for it anywhere.
geometry
edited Nov 29 '18 at 21:17
Blue
47.8k870152
asked Nov 29 '18 at 20:54
PeroPero
1207
$endgroup$
Given acute $triangle ABC$ with orthocenter $H$, prove $$|AH|,h_a+|BH|,h_b+|CH|,h_c=frac12left(a^2+b^2+c^2right)$$
I tried to do something with the perimeter, area, half-perimeter, but I can't solve it. I am not good at contest problems, and I just started practicing. I'll be glad if I get an answer or just guidelines. I couldn't seem to find this problem solved or guidelines for it anywhere.
geometry
geometry
edited Nov 29 '18 at 21:17
Blue
47.8k870152
asked Nov 29 '18 at 20:54
PeroPero
1207
edited Nov 29 '18 at 21:17
Blue
47.8k870152
asked Nov 29 '18 at 20:54
PeroPero
1207
edited Nov 29 '18 at 21:17
Blue
47.8k870152
edited Nov 29 '18 at 21:17
Blue
47.8k870152
edited Nov 29 '18 at 21:17
Blue
47.8k870152
47.8k870152
asked Nov 29 '18 at 20:54
PeroPero
1207
asked Nov 29 '18 at 20:54
PeroPero
1207
asked Nov 29 '18 at 20:54
PeroPero
1207
1207
closed as off-topic by T. Bongers, Théophile, KReiser, Alexander Gruber♦ Nov 30 '18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – T. Bongers, Théophile, KReiser, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by T. Bongers, Théophile, KReiser, Alexander Gruber♦ Nov 30 '18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – T. Bongers, Théophile, KReiser, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Welcome to MSE. What work have you done so far?
$endgroup$
– Théophile
Nov 29 '18 at 21:00
$begingroup$
Please define all objects that appear, starting with the triangle and $ha$ (which may be some $h_a$, typed ash_ain LaTeX), in the text, not (only) in the title. Then please show the own efforts to solve. Else there are some people that downvote from the start.
$endgroup$
– dan_fulea
Nov 29 '18 at 21:01
$begingroup$
If you don't want your question to get downvoted, you should include more context such as your own attempt at a solution. math.meta.stackexchange.com/questions/26017/…
$endgroup$
– Ryan Greyling
Nov 29 '18 at 21:02
$begingroup$
What are $h_a$, $h_b$, $h_c$? Are they the lengths of the altitudes from $A$, $B$, $C$?
$endgroup$
– Blue
Nov 29 '18 at 21:18
$begingroup$
I believe so. It wasn't specified in the problem.
$endgroup$
– Pero
Nov 29 '18 at 21:20
add a comment |
$begingroup$
Welcome to MSE. What work have you done so far?
$endgroup$
– Théophile
Nov 29 '18 at 21:00
$begingroup$
Please define all objects that appear, starting with the triangle and $ha$ (which may be some $h_a$, typed ash_ain LaTeX), in the text, not (only) in the title. Then please show the own efforts to solve. Else there are some people that downvote from the start.
$endgroup$
– dan_fulea
Nov 29 '18 at 21:01
$begingroup$
If you don't want your question to get downvoted, you should include more context such as your own attempt at a solution. math.meta.stackexchange.com/questions/26017/…
$endgroup$
– Ryan Greyling
Nov 29 '18 at 21:02
$begingroup$
What are $h_a$, $h_b$, $h_c$? Are they the lengths of the altitudes from $A$, $B$, $C$?
$endgroup$
– Blue
Nov 29 '18 at 21:18
$begingroup$
I believe so. It wasn't specified in the problem.
$endgroup$
– Pero
Nov 29 '18 at 21:20
$begingroup$
Welcome to MSE. What work have you done so far?
$endgroup$
– Théophile
Nov 29 '18 at 21:00
$begingroup$
Welcome to MSE. What work have you done so far?
$endgroup$
– Théophile
Nov 29 '18 at 21:00
$begingroup$
Please define all objects that appear, starting with the triangle and $ha$ (which may be some $h_a$, typed as
h_a in LaTeX), in the text, not (only) in the title. Then please show the own efforts to solve. Else there are some people that downvote from the start.$endgroup$
– dan_fulea
Nov 29 '18 at 21:01
$begingroup$
Please define all objects that appear, starting with the triangle and $ha$ (which may be some $h_a$, typed as
h_a in LaTeX), in the text, not (only) in the title. Then please show the own efforts to solve. Else there are some people that downvote from the start.$endgroup$
– dan_fulea
Nov 29 '18 at 21:01
$begingroup$
If you don't want your question to get downvoted, you should include more context such as your own attempt at a solution. math.meta.stackexchange.com/questions/26017/…
$endgroup$
– Ryan Greyling
Nov 29 '18 at 21:02
$begingroup$
If you don't want your question to get downvoted, you should include more context such as your own attempt at a solution. math.meta.stackexchange.com/questions/26017/…
$endgroup$
– Ryan Greyling
Nov 29 '18 at 21:02
$begingroup$
What are $h_a$, $h_b$, $h_c$? Are they the lengths of the altitudes from $A$, $B$, $C$?
$endgroup$
– Blue
Nov 29 '18 at 21:18
$begingroup$
What are $h_a$, $h_b$, $h_c$? Are they the lengths of the altitudes from $A$, $B$, $C$?
$endgroup$
– Blue
Nov 29 '18 at 21:18
$begingroup$
I believe so. It wasn't specified in the problem.
$endgroup$
– Pero
Nov 29 '18 at 21:20
$begingroup$
I believe so. It wasn't specified in the problem.
$endgroup$
– Pero
Nov 29 '18 at 21:20
add a comment |
1 Answer
1
active
oldest
votes
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If $A',B',C'$ are feet of the altitudes, from $ABA'sim AHC'$ we have $AHcdot h_a= ABcdot AC'$. Similarly, from $BAB'sim BHC'$, $BHcdot h_b= ABcdot BC'$ follows. By adding we have $AHcdot h_a+BHcdot h_b=AB(AC'+BC')= AB^2=c^2$. By symmetry, $BHcdot h_b+CHcdot h_c= a^2$ and $AHcdot h_a+CHcdot h_c= b^2$. Now just add these three equations.
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If $A',B',C'$ are feet of the altitudes, from $ABA'sim AHC'$ we have $AHcdot h_a= ABcdot AC'$. Similarly, from $BAB'sim BHC'$, $BHcdot h_b= ABcdot BC'$ follows. By adding we have $AHcdot h_a+BHcdot h_b=AB(AC'+BC')= AB^2=c^2$. By symmetry, $BHcdot h_b+CHcdot h_c= a^2$ and $AHcdot h_a+CHcdot h_c= b^2$. Now just add these three equations.
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
$endgroup$
add a comment |
$begingroup$
If $A',B',C'$ are feet of the altitudes, from $ABA'sim AHC'$ we have $AHcdot h_a= ABcdot AC'$. Similarly, from $BAB'sim BHC'$, $BHcdot h_b= ABcdot BC'$ follows. By adding we have $AHcdot h_a+BHcdot h_b=AB(AC'+BC')= AB^2=c^2$. By symmetry, $BHcdot h_b+CHcdot h_c= a^2$ and $AHcdot h_a+CHcdot h_c= b^2$. Now just add these three equations.
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
$endgroup$
add a comment |
$begingroup$
If $A',B',C'$ are feet of the altitudes, from $ABA'sim AHC'$ we have $AHcdot h_a= ABcdot AC'$. Similarly, from $BAB'sim BHC'$, $BHcdot h_b= ABcdot BC'$ follows. By adding we have $AHcdot h_a+BHcdot h_b=AB(AC'+BC')= AB^2=c^2$. By symmetry, $BHcdot h_b+CHcdot h_c= a^2$ and $AHcdot h_a+CHcdot h_c= b^2$. Now just add these three equations.
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
$endgroup$
If $A',B',C'$ are feet of the altitudes, from $ABA'sim AHC'$ we have $AHcdot h_a= ABcdot AC'$. Similarly, from $BAB'sim BHC'$, $BHcdot h_b= ABcdot BC'$ follows. By adding we have $AHcdot h_a+BHcdot h_b=AB(AC'+BC')= AB^2=c^2$. By symmetry, $BHcdot h_b+CHcdot h_c= a^2$ and $AHcdot h_a+CHcdot h_c= b^2$. Now just add these three equations.
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
answered Nov 29 '18 at 22:35
SMMSMM
2,19359
2,19359
add a comment |
add a comment |
$begingroup$
Welcome to MSE. What work have you done so far?
$endgroup$
– Théophile
Nov 29 '18 at 21:00
$begingroup$
Please define all objects that appear, starting with the triangle and $ha$ (which may be some $h_a$, typed as
h_ain LaTeX), in the text, not (only) in the title. Then please show the own efforts to solve. Else there are some people that downvote from the start.$endgroup$
– dan_fulea
Nov 29 '18 at 21:01
$begingroup$
If you don't want your question to get downvoted, you should include more context such as your own attempt at a solution. math.meta.stackexchange.com/questions/26017/…
$endgroup$
– Ryan Greyling
Nov 29 '18 at 21:02
$begingroup$
What are $h_a$, $h_b$, $h_c$? Are they the lengths of the altitudes from $A$, $B$, $C$?
$endgroup$
– Blue
Nov 29 '18 at 21:18
$begingroup$
I believe so. It wasn't specified in the problem.
$endgroup$
– Pero
Nov 29 '18 at 21:20