Internal angles in regular 18-gon
up vote
8
down vote
favorite
This (seemingly simple) problem is driving me nuts.
Find angle $alpha$ shown in the following regular 18-gon.
It was easy to find the angle between pink diagonals ($60^circ$). And I was able to solve the problem with some trigonometry (getting nice integer angle). However, all my attempts to solve the problem without use of trigonometry have failed. It looked like I was close to solution all the time (so many angles are equal to $60^circ$ or $120^circ$. I felt like I had to draw just one more line and the problem would break apart. I also tried with internal symmetries and rotations but eventually I had to give up.
Is there a way to solve this kind of problem without sines and cosines?
euclidean-geometry polygons plane-geometry
asked Dec 4 at 10:51
Oldboy
6,1481628
add a comment |
up vote
8
down vote
favorite
This (seemingly simple) problem is driving me nuts.
Find angle $alpha$ shown in the following regular 18-gon.
It was easy to find the angle between pink diagonals ($60^circ$). And I was able to solve the problem with some trigonometry (getting nice integer angle). However, all my attempts to solve the problem without use of trigonometry have failed. It looked like I was close to solution all the time (so many angles are equal to $60^circ$ or $120^circ$. I felt like I had to draw just one more line and the problem would break apart. I also tried with internal symmetries and rotations but eventually I had to give up.
Is there a way to solve this kind of problem without sines and cosines?
euclidean-geometry polygons plane-geometry
asked Dec 4 at 10:51
Oldboy
6,1481628
You should mention what the "nice integer angle" is, to save people some time.
– Blue
Dec 4 at 11:47
It's $20^circ$. Even more interesting is that if you extend the black line to the southeast it will pass through a vertex of the polygon.
– Oldboy
Dec 4 at 12:02
Say, the triangle with $alpha$ has vertices $A_1,A_2$ from the 18-gon. You claim the black line is $A_2A_8$. Once it's known, we can finish by observing $A_2A_8parallel A_3A_7$ and $A_1A_7A_3angle =20^circ$.
– Berci
Dec 7 at 8:25
add a comment |
up vote
8
down vote
favorite
up vote
8
down vote
favorite
This (seemingly simple) problem is driving me nuts.
Find angle $alpha$ shown in the following regular 18-gon.
It was easy to find the angle between pink diagonals ($60^circ$). And I was able to solve the problem with some trigonometry (getting nice integer angle). However, all my attempts to solve the problem without use of trigonometry have failed. It looked like I was close to solution all the time (so many angles are equal to $60^circ$ or $120^circ$. I felt like I had to draw just one more line and the problem would break apart. I also tried with internal symmetries and rotations but eventually I had to give up.
Is there a way to solve this kind of problem without sines and cosines?
euclidean-geometry polygons plane-geometry
asked Dec 4 at 10:51
Oldboy
6,1481628
This (seemingly simple) problem is driving me nuts.
Find angle $alpha$ shown in the following regular 18-gon.
It was easy to find the angle between pink diagonals ($60^circ$). And I was able to solve the problem with some trigonometry (getting nice integer angle). However, all my attempts to solve the problem without use of trigonometry have failed. It looked like I was close to solution all the time (so many angles are equal to $60^circ$ or $120^circ$. I felt like I had to draw just one more line and the problem would break apart. I also tried with internal symmetries and rotations but eventually I had to give up.
Is there a way to solve this kind of problem without sines and cosines?
euclidean-geometry polygons plane-geometry
euclidean-geometry polygons plane-geometry
asked Dec 4 at 10:51
Oldboy
6,1481628
asked Dec 4 at 10:51
Oldboy
6,1481628
asked Dec 4 at 10:51
Oldboy
6,1481628
asked Dec 4 at 10:51
Oldboy
6,1481628
asked Dec 4 at 10:51
Oldboy
6,1481628
6,1481628
You should mention what the "nice integer angle" is, to save people some time.
– Blue
Dec 4 at 11:47
It's $20^circ$. Even more interesting is that if you extend the black line to the southeast it will pass through a vertex of the polygon.
– Oldboy
Dec 4 at 12:02
Say, the triangle with $alpha$ has vertices $A_1,A_2$ from the 18-gon. You claim the black line is $A_2A_8$. Once it's known, we can finish by observing $A_2A_8parallel A_3A_7$ and $A_1A_7A_3angle =20^circ$.
– Berci
Dec 7 at 8:25
add a comment |
You should mention what the "nice integer angle" is, to save people some time.
– Blue
Dec 4 at 11:47
It's $20^circ$. Even more interesting is that if you extend the black line to the southeast it will pass through a vertex of the polygon.
– Oldboy
Dec 4 at 12:02
Say, the triangle with $alpha$ has vertices $A_1,A_2$ from the 18-gon. You claim the black line is $A_2A_8$. Once it's known, we can finish by observing $A_2A_8parallel A_3A_7$ and $A_1A_7A_3angle =20^circ$.
– Berci
Dec 7 at 8:25
You should mention what the "nice integer angle" is, to save people some time.
– Blue
Dec 4 at 11:47
You should mention what the "nice integer angle" is, to save people some time.
– Blue
Dec 4 at 11:47
It's $20^circ$. Even more interesting is that if you extend the black line to the southeast it will pass through a vertex of the polygon.
– Oldboy
Dec 4 at 12:02
It's $20^circ$. Even more interesting is that if you extend the black line to the southeast it will pass through a vertex of the polygon.
– Oldboy
Dec 4 at 12:02
Say, the triangle with $alpha$ has vertices $A_1,A_2$ from the 18-gon. You claim the black line is $A_2A_8$. Once it's known, we can finish by observing $A_2A_8parallel A_3A_7$ and $A_1A_7A_3angle =20^circ$.
– Berci
Dec 7 at 8:25
Say, the triangle with $alpha$ has vertices $A_1,A_2$ from the 18-gon. You claim the black line is $A_2A_8$. Once it's known, we can finish by observing $A_2A_8parallel A_3A_7$ and $A_1A_7A_3angle =20^circ$.
– Berci
Dec 7 at 8:25
add a comment |
active
oldest
votes
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',
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%2f3025421%2finternal-angles-in-regular-18-gon%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3025421%2finternal-angles-in-regular-18-gon%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
You should mention what the "nice integer angle" is, to save people some time.
– Blue
Dec 4 at 11:47
It's $20^circ$. Even more interesting is that if you extend the black line to the southeast it will pass through a vertex of the polygon.
– Oldboy
Dec 4 at 12:02
Say, the triangle with $alpha$ has vertices $A_1,A_2$ from the 18-gon. You claim the black line is $A_2A_8$. Once it's known, we can finish by observing $A_2A_8parallel A_3A_7$ and $A_1A_7A_3angle =20^circ$.
– Berci
Dec 7 at 8:25