Must inscribed rectangles share a center point with each other?
$begingroup$
If a rectangle is inscribed inside another rectangle, will they always share a center point? My intuition says yes, but I am not sure how to prove it to myself.
geometry euclidean-geometry
$endgroup$
add a comment |
$begingroup$
If a rectangle is inscribed inside another rectangle, will they always share a center point? My intuition says yes, but I am not sure how to prove it to myself.
geometry euclidean-geometry
$endgroup$
1
$begingroup$
Prove it by symmetry. Any rectangle rotated to any angular position about it's center point will have its diagonally opposite corners the same horizontal and vertical distances from the center.
$endgroup$
– Phil H
Dec 18 '18 at 22:54
$begingroup$
Have you checked that it is possible to inscribe a rectangle in a nonsquare rectangle? I'm not sure you can. If that's the case, then for a square the answer is clearly "yes". for a nonsquare it's "yes" but a vacuous assertion.
$endgroup$
– Ethan Bolker
Dec 18 '18 at 23:26
$begingroup$
@EthanBolker Why not? Imagine you already have the inner rectangle, through the opposite vertices, draw two pairs of parallel lines which are mutually perpendicular. I highly doubt you would get a square.
$endgroup$
– Quang Hoang
Dec 19 '18 at 3:01
$begingroup$
@QuangHoang Right you are.
$endgroup$
– Ethan Bolker
Dec 19 '18 at 14:33
add a comment |
$begingroup$
If a rectangle is inscribed inside another rectangle, will they always share a center point? My intuition says yes, but I am not sure how to prove it to myself.
geometry euclidean-geometry
$endgroup$
If a rectangle is inscribed inside another rectangle, will they always share a center point? My intuition says yes, but I am not sure how to prove it to myself.
geometry euclidean-geometry
geometry euclidean-geometry
edited Dec 18 '18 at 22:40
Mark
asked Dec 18 '18 at 22:32
MarkMark
132
132
1
$begingroup$
Prove it by symmetry. Any rectangle rotated to any angular position about it's center point will have its diagonally opposite corners the same horizontal and vertical distances from the center.
$endgroup$
– Phil H
Dec 18 '18 at 22:54
$begingroup$
Have you checked that it is possible to inscribe a rectangle in a nonsquare rectangle? I'm not sure you can. If that's the case, then for a square the answer is clearly "yes". for a nonsquare it's "yes" but a vacuous assertion.
$endgroup$
– Ethan Bolker
Dec 18 '18 at 23:26
$begingroup$
@EthanBolker Why not? Imagine you already have the inner rectangle, through the opposite vertices, draw two pairs of parallel lines which are mutually perpendicular. I highly doubt you would get a square.
$endgroup$
– Quang Hoang
Dec 19 '18 at 3:01
$begingroup$
@QuangHoang Right you are.
$endgroup$
– Ethan Bolker
Dec 19 '18 at 14:33
add a comment |
1
$begingroup$
Prove it by symmetry. Any rectangle rotated to any angular position about it's center point will have its diagonally opposite corners the same horizontal and vertical distances from the center.
$endgroup$
– Phil H
Dec 18 '18 at 22:54
$begingroup$
Have you checked that it is possible to inscribe a rectangle in a nonsquare rectangle? I'm not sure you can. If that's the case, then for a square the answer is clearly "yes". for a nonsquare it's "yes" but a vacuous assertion.
$endgroup$
– Ethan Bolker
Dec 18 '18 at 23:26
$begingroup$
@EthanBolker Why not? Imagine you already have the inner rectangle, through the opposite vertices, draw two pairs of parallel lines which are mutually perpendicular. I highly doubt you would get a square.
$endgroup$
– Quang Hoang
Dec 19 '18 at 3:01
$begingroup$
@QuangHoang Right you are.
$endgroup$
– Ethan Bolker
Dec 19 '18 at 14:33
1
1
$begingroup$
Prove it by symmetry. Any rectangle rotated to any angular position about it's center point will have its diagonally opposite corners the same horizontal and vertical distances from the center.
$endgroup$
– Phil H
Dec 18 '18 at 22:54
$begingroup$
Prove it by symmetry. Any rectangle rotated to any angular position about it's center point will have its diagonally opposite corners the same horizontal and vertical distances from the center.
$endgroup$
– Phil H
Dec 18 '18 at 22:54
$begingroup$
Have you checked that it is possible to inscribe a rectangle in a nonsquare rectangle? I'm not sure you can. If that's the case, then for a square the answer is clearly "yes". for a nonsquare it's "yes" but a vacuous assertion.
$endgroup$
– Ethan Bolker
Dec 18 '18 at 23:26
$begingroup$
Have you checked that it is possible to inscribe a rectangle in a nonsquare rectangle? I'm not sure you can. If that's the case, then for a square the answer is clearly "yes". for a nonsquare it's "yes" but a vacuous assertion.
$endgroup$
– Ethan Bolker
Dec 18 '18 at 23:26
$begingroup$
@EthanBolker Why not? Imagine you already have the inner rectangle, through the opposite vertices, draw two pairs of parallel lines which are mutually perpendicular. I highly doubt you would get a square.
$endgroup$
– Quang Hoang
Dec 19 '18 at 3:01
$begingroup$
@EthanBolker Why not? Imagine you already have the inner rectangle, through the opposite vertices, draw two pairs of parallel lines which are mutually perpendicular. I highly doubt you would get a square.
$endgroup$
– Quang Hoang
Dec 19 '18 at 3:01
$begingroup$
@QuangHoang Right you are.
$endgroup$
– Ethan Bolker
Dec 19 '18 at 14:33
$begingroup$
@QuangHoang Right you are.
$endgroup$
– Ethan Bolker
Dec 19 '18 at 14:33
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: $angle FAB = angle HCD$ as they have (pairwise) parallel sides. So $triangle FAB = triangle HCD$. And we conclude that $FH$ intersects $BD$ at their common midpoints.
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3045790%2fmust-inscribed-rectangles-share-a-center-point-with-each-other%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: $angle FAB = angle HCD$ as they have (pairwise) parallel sides. So $triangle FAB = triangle HCD$. And we conclude that $FH$ intersects $BD$ at their common midpoints.
$endgroup$
add a comment |
$begingroup$
Hint: $angle FAB = angle HCD$ as they have (pairwise) parallel sides. So $triangle FAB = triangle HCD$. And we conclude that $FH$ intersects $BD$ at their common midpoints.
$endgroup$
add a comment |
$begingroup$
Hint: $angle FAB = angle HCD$ as they have (pairwise) parallel sides. So $triangle FAB = triangle HCD$. And we conclude that $FH$ intersects $BD$ at their common midpoints.
$endgroup$
Hint: $angle FAB = angle HCD$ as they have (pairwise) parallel sides. So $triangle FAB = triangle HCD$. And we conclude that $FH$ intersects $BD$ at their common midpoints.
answered Dec 19 '18 at 3:08
Quang HoangQuang Hoang
13.2k1233
13.2k1233
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3045790%2fmust-inscribed-rectangles-share-a-center-point-with-each-other%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
Prove it by symmetry. Any rectangle rotated to any angular position about it's center point will have its diagonally opposite corners the same horizontal and vertical distances from the center.
$endgroup$
– Phil H
Dec 18 '18 at 22:54
$begingroup$
Have you checked that it is possible to inscribe a rectangle in a nonsquare rectangle? I'm not sure you can. If that's the case, then for a square the answer is clearly "yes". for a nonsquare it's "yes" but a vacuous assertion.
$endgroup$
– Ethan Bolker
Dec 18 '18 at 23:26
$begingroup$
@EthanBolker Why not? Imagine you already have the inner rectangle, through the opposite vertices, draw two pairs of parallel lines which are mutually perpendicular. I highly doubt you would get a square.
$endgroup$
– Quang Hoang
Dec 19 '18 at 3:01
$begingroup$
@QuangHoang Right you are.
$endgroup$
– Ethan Bolker
Dec 19 '18 at 14:33