Let $H$ be the orthocentre of triangle $ABC$. Prove that the midpoints of $AB, AC, AH, BC, BH$ and $CH$ form...
$begingroup$
Let $H$ be the orthocentre of triangle $ABC$ ($H$ is the point inside triangle $ABC$ such that $AH ⊥ BC$, and $BH ⊥ AC$ and $CH ⊥ AB$). Prove that the midpoints of $AB, AC, AH, BC, BH$ and $CH$ form a cyclic hexagon.
geometry
edited Jan 19 at 8:44
idriskameni
644319
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
$endgroup$
add a comment |
$begingroup$
Let $H$ be the orthocentre of triangle $ABC$ ($H$ is the point inside triangle $ABC$ such that $AH ⊥ BC$, and $BH ⊥ AC$ and $CH ⊥ AB$). Prove that the midpoints of $AB, AC, AH, BC, BH$ and $CH$ form a cyclic hexagon.
geometry
edited Jan 19 at 8:44
idriskameni
644319
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
$endgroup$
$begingroup$
You have posted six questions in the past day or so, each consisting of an isolated question. Typically, the community prefers questions that include some idea of what you know about the problem (what you've tried, why you're interested, where the problem comes from, what tools are allowed, etc) and/or where exactly you got stuck. Such information helps answerers target their responses to your skill level or specific difficulty, without wasting time telling you things you already know. (It also helps convince people that you aren't simply trying to get them to do your homework for you.)
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– Blue
Jan 19 at 7:25
add a comment |
$begingroup$
Let $H$ be the orthocentre of triangle $ABC$ ($H$ is the point inside triangle $ABC$ such that $AH ⊥ BC$, and $BH ⊥ AC$ and $CH ⊥ AB$). Prove that the midpoints of $AB, AC, AH, BC, BH$ and $CH$ form a cyclic hexagon.
geometry
edited Jan 19 at 8:44
idriskameni
644319
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
$endgroup$
Let $H$ be the orthocentre of triangle $ABC$ ($H$ is the point inside triangle $ABC$ such that $AH ⊥ BC$, and $BH ⊥ AC$ and $CH ⊥ AB$). Prove that the midpoints of $AB, AC, AH, BC, BH$ and $CH$ form a cyclic hexagon.
geometry
geometry
edited Jan 19 at 8:44
idriskameni
644319
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
edited Jan 19 at 8:44
idriskameni
644319
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
edited Jan 19 at 8:44
idriskameni
644319
edited Jan 19 at 8:44
idriskameni
644319
edited Jan 19 at 8:44
idriskameni
644319
644319
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
asked Jan 19 at 7:10
Anson ChanAnson Chan
162
162
$begingroup$
You have posted six questions in the past day or so, each consisting of an isolated question. Typically, the community prefers questions that include some idea of what you know about the problem (what you've tried, why you're interested, where the problem comes from, what tools are allowed, etc) and/or where exactly you got stuck. Such information helps answerers target their responses to your skill level or specific difficulty, without wasting time telling you things you already know. (It also helps convince people that you aren't simply trying to get them to do your homework for you.)
$endgroup$
– Blue
Jan 19 at 7:25
add a comment |
$begingroup$
You have posted six questions in the past day or so, each consisting of an isolated question. Typically, the community prefers questions that include some idea of what you know about the problem (what you've tried, why you're interested, where the problem comes from, what tools are allowed, etc) and/or where exactly you got stuck. Such information helps answerers target their responses to your skill level or specific difficulty, without wasting time telling you things you already know. (It also helps convince people that you aren't simply trying to get them to do your homework for you.)
$endgroup$
– Blue
Jan 19 at 7:25
$begingroup$
You have posted six questions in the past day or so, each consisting of an isolated question. Typically, the community prefers questions that include some idea of what you know about the problem (what you've tried, why you're interested, where the problem comes from, what tools are allowed, etc) and/or where exactly you got stuck. Such information helps answerers target their responses to your skill level or specific difficulty, without wasting time telling you things you already know. (It also helps convince people that you aren't simply trying to get them to do your homework for you.)
$endgroup$
– Blue
Jan 19 at 7:25
$begingroup$
You have posted six questions in the past day or so, each consisting of an isolated question. Typically, the community prefers questions that include some idea of what you know about the problem (what you've tried, why you're interested, where the problem comes from, what tools are allowed, etc) and/or where exactly you got stuck. Such information helps answerers target their responses to your skill level or specific difficulty, without wasting time telling you things you already know. (It also helps convince people that you aren't simply trying to get them to do your homework for you.)
$endgroup$
– Blue
Jan 19 at 7:25
add a comment |
1 Answer
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$begingroup$
The circumcirle of this hexagon it's the nine-point circle.
See here: https://en.wikipedia.org/wiki/Nine-point_circle
Also, the following can help.
https://en.wikipedia.org/wiki/Trillium_theorem
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The circumcirle of this hexagon it's the nine-point circle.
See here: https://en.wikipedia.org/wiki/Nine-point_circle
Also, the following can help.
https://en.wikipedia.org/wiki/Trillium_theorem
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
$endgroup$
add a comment |
$begingroup$
The circumcirle of this hexagon it's the nine-point circle.
See here: https://en.wikipedia.org/wiki/Nine-point_circle
Also, the following can help.
https://en.wikipedia.org/wiki/Trillium_theorem
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
$endgroup$
add a comment |
$begingroup$
The circumcirle of this hexagon it's the nine-point circle.
See here: https://en.wikipedia.org/wiki/Nine-point_circle
Also, the following can help.
https://en.wikipedia.org/wiki/Trillium_theorem
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
$endgroup$
The circumcirle of this hexagon it's the nine-point circle.
See here: https://en.wikipedia.org/wiki/Nine-point_circle
Also, the following can help.
https://en.wikipedia.org/wiki/Trillium_theorem
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
edited Jan 19 at 7:25
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
answered Jan 19 at 7:19
Michael RozenbergMichael Rozenberg
103k1891195
103k1891195
add a comment |
add a comment |
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You have posted six questions in the past day or so, each consisting of an isolated question. Typically, the community prefers questions that include some idea of what you know about the problem (what you've tried, why you're interested, where the problem comes from, what tools are allowed, etc) and/or where exactly you got stuck. Such information helps answerers target their responses to your skill level or specific difficulty, without wasting time telling you things you already know. (It also helps convince people that you aren't simply trying to get them to do your homework for you.)
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– Blue
Jan 19 at 7:25