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 circles triangles
edited Jan 8 at 3:20
Namaste
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 circles triangles
edited Jan 8 at 3:20
Namaste
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 circles triangles
edited Jan 8 at 3:20
Namaste
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 circles triangles
geometry contest-math euclidean-geometry circles triangles
edited Jan 8 at 3:20
Namaste
1
asked Jul 3 '18 at 5:02
user573736user573736
61
edited Jan 8 at 3:20
Namaste
1
asked Jul 3 '18 at 5:02
user573736user573736
61
edited Jan 8 at 3:20
Namaste
1
edited Jan 8 at 3:20
Namaste
1
edited Jan 8 at 3:20
Namaste
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
111k1897201
$endgroup$
$begingroup$
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
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1 Answer
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oldest
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1 Answer
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oldest
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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
111k1897201
$endgroup$
$begingroup$
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
111k1897201
$endgroup$
$begingroup$
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
111k1897201
$endgroup$
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
111k1897201
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
111k1897201
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
111k1897201
answered Jul 3 '18 at 5:26
Michael RozenbergMichael Rozenberg
111k1897201
111k1897201
$begingroup$
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
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