concurrency of angle bisectors, medians, perpendicular bisectors, altitudes
$begingroup$
When you think of a triangle the basic constructions you think of are perpendicular bisectors of the sides, angle bisectors, altitudes, and medians. Now if you were someone who just started learning about these concepts it would seem very unusual that all these basic constructions related to the triangle are concurrent. To me that seems really unlikely and I ask, is there some deep lying meaning or reason as to why this is so? Any help would be appreciated.
NOTE : I am not asking for the proofs of the concurrency theorems
geometry
asked May 10 '15 at 4:56
user34304user34304
1,29511330
$endgroup$
add a comment |
$begingroup$
When you think of a triangle the basic constructions you think of are perpendicular bisectors of the sides, angle bisectors, altitudes, and medians. Now if you were someone who just started learning about these concepts it would seem very unusual that all these basic constructions related to the triangle are concurrent. To me that seems really unlikely and I ask, is there some deep lying meaning or reason as to why this is so? Any help would be appreciated.
NOTE : I am not asking for the proofs of the concurrency theorems
geometry
asked May 10 '15 at 4:56
user34304user34304
1,29511330
$endgroup$
add a comment |
$begingroup$
When you think of a triangle the basic constructions you think of are perpendicular bisectors of the sides, angle bisectors, altitudes, and medians. Now if you were someone who just started learning about these concepts it would seem very unusual that all these basic constructions related to the triangle are concurrent. To me that seems really unlikely and I ask, is there some deep lying meaning or reason as to why this is so? Any help would be appreciated.
NOTE : I am not asking for the proofs of the concurrency theorems
geometry
asked May 10 '15 at 4:56
user34304user34304
1,29511330
$endgroup$
When you think of a triangle the basic constructions you think of are perpendicular bisectors of the sides, angle bisectors, altitudes, and medians. Now if you were someone who just started learning about these concepts it would seem very unusual that all these basic constructions related to the triangle are concurrent. To me that seems really unlikely and I ask, is there some deep lying meaning or reason as to why this is so? Any help would be appreciated.
NOTE : I am not asking for the proofs of the concurrency theorems
geometry
geometry
asked May 10 '15 at 4:56
user34304user34304
1,29511330
asked May 10 '15 at 4:56
user34304user34304
1,29511330
asked May 10 '15 at 4:56
user34304user34304
1,29511330
asked May 10 '15 at 4:56
user34304user34304
1,29511330
asked May 10 '15 at 4:56
user34304user34304
1,29511330
1,29511330
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The concept of concurrencies of various centers of a triangle can be appreciated from 3 different levels.
At the junior level, the said constructions are mainly done by the help of RULERs and PROTRACTORs. The point is to arouse interest and see if lines can be accurately or not.
At the intermediate level, lines should be drawn using STRAIGHT-EDGE and COMPASSES only. Techniques required are preliminary knowledge of congruency. Thus, the learning tasks are scheduled as studying congruency first and the topic of constructions right after it. The construction exercise can therefore also be considered as a consolidation of the topics on congruency.
At the advanced level, the truth of the concurrencies should be proved formally.
answered May 10 '15 at 9:16
MickMick
12k31641
$endgroup$
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%2f1275134%2fconcurrency-of-angle-bisectors-medians-perpendicular-bisectors-altitudes%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
$begingroup$
The concept of concurrencies of various centers of a triangle can be appreciated from 3 different levels.
At the junior level, the said constructions are mainly done by the help of RULERs and PROTRACTORs. The point is to arouse interest and see if lines can be accurately or not.
At the intermediate level, lines should be drawn using STRAIGHT-EDGE and COMPASSES only. Techniques required are preliminary knowledge of congruency. Thus, the learning tasks are scheduled as studying congruency first and the topic of constructions right after it. The construction exercise can therefore also be considered as a consolidation of the topics on congruency.
At the advanced level, the truth of the concurrencies should be proved formally.
answered May 10 '15 at 9:16
MickMick
12k31641
$endgroup$
add a comment |
$begingroup$
The concept of concurrencies of various centers of a triangle can be appreciated from 3 different levels.
At the junior level, the said constructions are mainly done by the help of RULERs and PROTRACTORs. The point is to arouse interest and see if lines can be accurately or not.
At the intermediate level, lines should be drawn using STRAIGHT-EDGE and COMPASSES only. Techniques required are preliminary knowledge of congruency. Thus, the learning tasks are scheduled as studying congruency first and the topic of constructions right after it. The construction exercise can therefore also be considered as a consolidation of the topics on congruency.
At the advanced level, the truth of the concurrencies should be proved formally.
answered May 10 '15 at 9:16
MickMick
12k31641
$endgroup$
add a comment |
$begingroup$
The concept of concurrencies of various centers of a triangle can be appreciated from 3 different levels.
At the junior level, the said constructions are mainly done by the help of RULERs and PROTRACTORs. The point is to arouse interest and see if lines can be accurately or not.
At the intermediate level, lines should be drawn using STRAIGHT-EDGE and COMPASSES only. Techniques required are preliminary knowledge of congruency. Thus, the learning tasks are scheduled as studying congruency first and the topic of constructions right after it. The construction exercise can therefore also be considered as a consolidation of the topics on congruency.
At the advanced level, the truth of the concurrencies should be proved formally.
answered May 10 '15 at 9:16
MickMick
12k31641
$endgroup$
The concept of concurrencies of various centers of a triangle can be appreciated from 3 different levels.
At the junior level, the said constructions are mainly done by the help of RULERs and PROTRACTORs. The point is to arouse interest and see if lines can be accurately or not.
At the intermediate level, lines should be drawn using STRAIGHT-EDGE and COMPASSES only. Techniques required are preliminary knowledge of congruency. Thus, the learning tasks are scheduled as studying congruency first and the topic of constructions right after it. The construction exercise can therefore also be considered as a consolidation of the topics on congruency.
At the advanced level, the truth of the concurrencies should be proved formally.
answered May 10 '15 at 9:16
MickMick
12k31641
edited Jan 5 at 19:00
answered May 10 '15 at 9:16
MickMick
12k31641
answered May 10 '15 at 9:16
MickMick
12k31641
answered May 10 '15 at 9:16
MickMick
12k31641
12k31641
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%2f1275134%2fconcurrency-of-angle-bisectors-medians-perpendicular-bisectors-altitudes%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
