Mixing
What are the values of A?
$begingroup$
Question- Triangle $ABC$ has incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BXcdot AB = IB^2$ and $CYcdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$?
I have solved it and my answer is $60^{circ}$. Are there any other possible values of $A$, because the question says "possible values of $A$"? Have I missed anything or any other values of $A$?
Please help. And this question is from the Indian National Mathematics Olympiad - 1991.
geometry contest-math euclidean-geometry circle triangle
edited Jan 8 at 3:20
amWhy
1
asked Jul 3 '18 at 5:02
user573736user573736
61
$endgroup$
add a comment |
$begingroup$
Question- Triangle $ABC$ has incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BXcdot AB = IB^2$ and $CYcdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$?
I have solved it and my answer is $60^{circ}$. Are there any other possible values of $A$, because the question says "possible values of $A$"? Have I missed anything or any other values of $A$?
Please help. And this question is from the Indian National Mathematics Olympiad - 1991.
geometry contest-math euclidean-geometry circle triangle
edited Jan 8 at 3:20
amWhy
1
asked Jul 3 '18 at 5:02
user573736user573736
61
$endgroup$
add a comment |
$begingroup$
Question- Triangle $ABC$ has incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BXcdot AB = IB^2$ and $CYcdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$?
I have solved it and my answer is $60^{circ}$. Are there any other possible values of $A$, because the question says "possible values of $A$"? Have I missed anything or any other values of $A$?
Please help. And this question is from the Indian National Mathematics Olympiad - 1991.
geometry contest-math euclidean-geometry circle triangle
edited Jan 8 at 3:20
amWhy
1
asked Jul 3 '18 at 5:02
user573736user573736
61
$endgroup$
Question- Triangle $ABC$ has incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BXcdot AB = IB^2$ and $CYcdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$?
I have solved it and my answer is $60^{circ}$. Are there any other possible values of $A$, because the question says "possible values of $A$"? Have I missed anything or any other values of $A$?
Please help. And this question is from the Indian National Mathematics Olympiad - 1991.
geometry contest-math euclidean-geometry circle triangle
geometry contest-math euclidean-geometry circle triangle
edited Jan 8 at 3:20
amWhy
1
asked Jul 3 '18 at 5:02
user573736user573736
61
edited Jan 8 at 3:20
amWhy
1
asked Jul 3 '18 at 5:02
user573736user573736
61
edited Jan 8 at 3:20
amWhy
1
edited Jan 8 at 3:20
amWhy
1
edited Jan 8 at 3:20
amWhy
1
1
asked Jul 3 '18 at 5:02
user573736user573736
61
asked Jul 3 '18 at 5:02
user573736user573736
61
asked Jul 3 '18 at 5:02
user573736user573736
61
61
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
By the given $BI$ is a tangent line to the circle $(AIX)$, which gives $measuredangle XIB=frac{alpha}{2}$.
Similarly, $measuredangle YIC=frac{alpha}{2}.$
Thus, since $X$, $Y$ and $I$ are collinear, we obtain: $$alpha=frac{beta}{2}+frac{gamma}{2}$$ or
$$2alpha=180^{circ}-alpha$$ or $$alpha=60^{circ}.$$
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
$endgroup$
$begingroup$
So there is only one possible value, i.e. 60°?
$endgroup$
– user573736
Jul 3 '18 at 9:05
$begingroup$
Yes, of course!
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 15:39
$begingroup$
One More Question Please tell... Question- Two circles C1 & C2 of radii a and b touch each other externally and they also touch a unit circle C internally. A circle C3 of radius r is inscribed to touch the circle C1 & C2 externally and the circle C internally. Find r in terms of a and b. Answer - r=ab1−ab. I have posted it in my profile, but no one responded...
$endgroup$
– user573736
Jul 3 '18 at 15:44
$begingroup$
I think it happens because you do not accept answers.
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 16:21
$begingroup$
What? I don't accept answers! How to accept answers?
$endgroup$
– user573736
Jul 3 '18 at 16:49
|
show 2 more comments
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%2f2839239%2fwhat-are-the-values-of-a%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
$begingroup$
By the given $BI$ is a tangent line to the circle $(AIX)$, which gives $measuredangle XIB=frac{alpha}{2}$.
Similarly, $measuredangle YIC=frac{alpha}{2}.$
Thus, since $X$, $Y$ and $I$ are collinear, we obtain: $$alpha=frac{beta}{2}+frac{gamma}{2}$$ or
$$2alpha=180^{circ}-alpha$$ or $$alpha=60^{circ}.$$
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
$endgroup$
$begingroup$
So there is only one possible value, i.e. 60°?
$endgroup$
– user573736
Jul 3 '18 at 9:05
$begingroup$
Yes, of course!
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 15:39
$begingroup$
One More Question Please tell... Question- Two circles C1 & C2 of radii a and b touch each other externally and they also touch a unit circle C internally. A circle C3 of radius r is inscribed to touch the circle C1 & C2 externally and the circle C internally. Find r in terms of a and b. Answer - r=ab1−ab. I have posted it in my profile, but no one responded...
$endgroup$
– user573736
Jul 3 '18 at 15:44
$begingroup$
I think it happens because you do not accept answers.
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 16:21
$begingroup$
What? I don't accept answers! How to accept answers?
$endgroup$
– user573736
Jul 3 '18 at 16:49
|
show 2 more comments
$begingroup$
By the given $BI$ is a tangent line to the circle $(AIX)$, which gives $measuredangle XIB=frac{alpha}{2}$.
Similarly, $measuredangle YIC=frac{alpha}{2}.$
Thus, since $X$, $Y$ and $I$ are collinear, we obtain: $$alpha=frac{beta}{2}+frac{gamma}{2}$$ or
$$2alpha=180^{circ}-alpha$$ or $$alpha=60^{circ}.$$
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
$endgroup$
$begingroup$
So there is only one possible value, i.e. 60°?
$endgroup$
– user573736
Jul 3 '18 at 9:05
$begingroup$
Yes, of course!
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 15:39
$begingroup$
One More Question Please tell... Question- Two circles C1 & C2 of radii a and b touch each other externally and they also touch a unit circle C internally. A circle C3 of radius r is inscribed to touch the circle C1 & C2 externally and the circle C internally. Find r in terms of a and b. Answer - r=ab1−ab. I have posted it in my profile, but no one responded...
$endgroup$
– user573736
Jul 3 '18 at 15:44
$begingroup$
I think it happens because you do not accept answers.
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 16:21
$begingroup$
What? I don't accept answers! How to accept answers?
$endgroup$
– user573736
Jul 3 '18 at 16:49
|
show 2 more comments
$begingroup$
By the given $BI$ is a tangent line to the circle $(AIX)$, which gives $measuredangle XIB=frac{alpha}{2}$.
Similarly, $measuredangle YIC=frac{alpha}{2}.$
Thus, since $X$, $Y$ and $I$ are collinear, we obtain: $$alpha=frac{beta}{2}+frac{gamma}{2}$$ or
$$2alpha=180^{circ}-alpha$$ or $$alpha=60^{circ}.$$
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
$endgroup$
By the given $BI$ is a tangent line to the circle $(AIX)$, which gives $measuredangle XIB=frac{alpha}{2}$.
Similarly, $measuredangle YIC=frac{alpha}{2}.$
Thus, since $X$, $Y$ and $I$ are collinear, we obtain: $$alpha=frac{beta}{2}+frac{gamma}{2}$$ or
$$2alpha=180^{circ}-alpha$$ or $$alpha=60^{circ}.$$
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
100k1591192
100k1591192
$begingroup$
So there is only one possible value, i.e. 60°?
$endgroup$
– user573736
Jul 3 '18 at 9:05
$begingroup$
Yes, of course!
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 15:39
$begingroup$
One More Question Please tell... Question- Two circles C1 & C2 of radii a and b touch each other externally and they also touch a unit circle C internally. A circle C3 of radius r is inscribed to touch the circle C1 & C2 externally and the circle C internally. Find r in terms of a and b. Answer - r=ab1−ab. I have posted it in my profile, but no one responded...
$endgroup$
– user573736
Jul 3 '18 at 15:44
$begingroup$
I think it happens because you do not accept answers.
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 16:21
$begingroup$
What? I don't accept answers! How to accept answers?
$endgroup$
– user573736
Jul 3 '18 at 16:49
|
show 2 more comments
$begingroup$
So there is only one possible value, i.e. 60°?
$endgroup$
– user573736
Jul 3 '18 at 9:05
$begingroup$
Yes, of course!
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 15:39
$begingroup$
One More Question Please tell... Question- Two circles C1 & C2 of radii a and b touch each other externally and they also touch a unit circle C internally. A circle C3 of radius r is inscribed to touch the circle C1 & C2 externally and the circle C internally. Find r in terms of a and b. Answer - r=ab1−ab. I have posted it in my profile, but no one responded...
$endgroup$
– user573736
Jul 3 '18 at 15:44
$begingroup$
I think it happens because you do not accept answers.
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 16:21
$begingroup$
What? I don't accept answers! How to accept answers?
$endgroup$
– user573736
Jul 3 '18 at 16:49
$begingroup$
So there is only one possible value, i.e. 60°?
$endgroup$
– user573736
Jul 3 '18 at 9:05
$begingroup$
So there is only one possible value, i.e. 60°?
$endgroup$
– user573736
Jul 3 '18 at 9:05
$begingroup$
Yes, of course!
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 15:39
$begingroup$
Yes, of course!
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 15:39
$begingroup$
One More Question Please tell... Question- Two circles C1 & C2 of radii a and b touch each other externally and they also touch a unit circle C internally. A circle C3 of radius r is inscribed to touch the circle C1 & C2 externally and the circle C internally. Find r in terms of a and b. Answer - r=ab1−ab. I have posted it in my profile, but no one responded...
$endgroup$
– user573736
Jul 3 '18 at 15:44
$begingroup$
One More Question Please tell... Question- Two circles C1 & C2 of radii a and b touch each other externally and they also touch a unit circle C internally. A circle C3 of radius r is inscribed to touch the circle C1 & C2 externally and the circle C internally. Find r in terms of a and b. Answer - r=ab1−ab. I have posted it in my profile, but no one responded...
$endgroup$
– user573736
Jul 3 '18 at 15:44
$begingroup$
I think it happens because you do not accept answers.
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 16:21
$begingroup$
I think it happens because you do not accept answers.
$endgroup$
– Michael Rozenberg
Jul 3 '18 at 16:21
$begingroup$
What? I don't accept answers! How to accept answers?
$endgroup$
– user573736
Jul 3 '18 at 16:49
$begingroup$
What? I don't accept answers! How to accept answers?
$endgroup$
– user573736
Jul 3 '18 at 16:49
|
show 2 more comments
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%2f2839239%2fwhat-are-the-values-of-a%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
