Ratio of radii product to radii sum of three touching circles
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Three circles touch one another externally.The tangents at their points of contact meet a point whose distance from a point of contact is 4.How to find ratio of product of the radii to the sum of radii of the circle?
I assumed the general 2nd degree equations of the three circles as $S_1,S_2$ and $S_3$.Found the three common tangent equations.How to proceed from there?Thanks.
geometry analytic-geometry circle
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add a comment |
$begingroup$
Three circles touch one another externally.The tangents at their points of contact meet a point whose distance from a point of contact is 4.How to find ratio of product of the radii to the sum of radii of the circle?
I assumed the general 2nd degree equations of the three circles as $S_1,S_2$ and $S_3$.Found the three common tangent equations.How to proceed from there?Thanks.
geometry analytic-geometry circle
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$begingroup$
Suppose the lines make angles $alpha$, $beta$, $gamma$ (with $alpha + beta + gamma = 2pi$). Express the radii in terms of trig functions of $alpha/2$, etc, (and the distance $4$), and simplify.
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– Blue
Sep 15 '15 at 20:01
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@Blue I'm not being able to simplify.Please write out atleast a few steps..
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– user220382
Sep 15 '15 at 21:37
add a comment |
$begingroup$
Three circles touch one another externally.The tangents at their points of contact meet a point whose distance from a point of contact is 4.How to find ratio of product of the radii to the sum of radii of the circle?
I assumed the general 2nd degree equations of the three circles as $S_1,S_2$ and $S_3$.Found the three common tangent equations.How to proceed from there?Thanks.
geometry analytic-geometry circle
$endgroup$
Three circles touch one another externally.The tangents at their points of contact meet a point whose distance from a point of contact is 4.How to find ratio of product of the radii to the sum of radii of the circle?
I assumed the general 2nd degree equations of the three circles as $S_1,S_2$ and $S_3$.Found the three common tangent equations.How to proceed from there?Thanks.
geometry analytic-geometry circle
geometry analytic-geometry circle
asked Sep 15 '15 at 19:11
user220382
$begingroup$
Suppose the lines make angles $alpha$, $beta$, $gamma$ (with $alpha + beta + gamma = 2pi$). Express the radii in terms of trig functions of $alpha/2$, etc, (and the distance $4$), and simplify.
$endgroup$
– Blue
Sep 15 '15 at 20:01
$begingroup$
@Blue I'm not being able to simplify.Please write out atleast a few steps..
$endgroup$
– user220382
Sep 15 '15 at 21:37
add a comment |
$begingroup$
Suppose the lines make angles $alpha$, $beta$, $gamma$ (with $alpha + beta + gamma = 2pi$). Express the radii in terms of trig functions of $alpha/2$, etc, (and the distance $4$), and simplify.
$endgroup$
– Blue
Sep 15 '15 at 20:01
$begingroup$
@Blue I'm not being able to simplify.Please write out atleast a few steps..
$endgroup$
– user220382
Sep 15 '15 at 21:37
$begingroup$
Suppose the lines make angles $alpha$, $beta$, $gamma$ (with $alpha + beta + gamma = 2pi$). Express the radii in terms of trig functions of $alpha/2$, etc, (and the distance $4$), and simplify.
$endgroup$
– Blue
Sep 15 '15 at 20:01
$begingroup$
Suppose the lines make angles $alpha$, $beta$, $gamma$ (with $alpha + beta + gamma = 2pi$). Express the radii in terms of trig functions of $alpha/2$, etc, (and the distance $4$), and simplify.
$endgroup$
– Blue
Sep 15 '15 at 20:01
$begingroup$
@Blue I'm not being able to simplify.Please write out atleast a few steps..
$endgroup$
– user220382
Sep 15 '15 at 21:37
$begingroup$
@Blue I'm not being able to simplify.Please write out atleast a few steps..
$endgroup$
– user220382
Sep 15 '15 at 21:37
add a comment |
1 Answer
1
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If the angles made by the lines at their common point are $2alpha$, $2beta$, $2 gamma$ (where their sum is $2pi$), then the three radii are pretty clearly
$$d tan alpha qquad dtan beta qquad d tan gamma$$
where $d$ is the distance from the point of intersection to any of the points of tangency. (In the given problem, $d=4$.) The product of the radii is trivial; as for the sum ...
Since
$$tanalpha = tan(pi-beta-gamma) = -tan(beta+gamma) = -frac{tanbeta+tangamma}{1-tanbetatangamma}$$
we have
$$tanalpha + tanbeta + tangamma = tanalpha-tanalpha(1-tanbetatangamma) = tanalphatanbetatangamma$$
Therefore,
$$frac{text{product of radii}}{text{sum of radii}} = frac{d^3tanalphatanbetatangamma}{dtanalphatanbetatangamma} = d^2$$
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
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1 Answer
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1 Answer
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active
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$begingroup$
If the angles made by the lines at their common point are $2alpha$, $2beta$, $2 gamma$ (where their sum is $2pi$), then the three radii are pretty clearly
$$d tan alpha qquad dtan beta qquad d tan gamma$$
where $d$ is the distance from the point of intersection to any of the points of tangency. (In the given problem, $d=4$.) The product of the radii is trivial; as for the sum ...
Since
$$tanalpha = tan(pi-beta-gamma) = -tan(beta+gamma) = -frac{tanbeta+tangamma}{1-tanbetatangamma}$$
we have
$$tanalpha + tanbeta + tangamma = tanalpha-tanalpha(1-tanbetatangamma) = tanalphatanbetatangamma$$
Therefore,
$$frac{text{product of radii}}{text{sum of radii}} = frac{d^3tanalphatanbetatangamma}{dtanalphatanbetatangamma} = d^2$$
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
$endgroup$
add a comment |
$begingroup$
If the angles made by the lines at their common point are $2alpha$, $2beta$, $2 gamma$ (where their sum is $2pi$), then the three radii are pretty clearly
$$d tan alpha qquad dtan beta qquad d tan gamma$$
where $d$ is the distance from the point of intersection to any of the points of tangency. (In the given problem, $d=4$.) The product of the radii is trivial; as for the sum ...
Since
$$tanalpha = tan(pi-beta-gamma) = -tan(beta+gamma) = -frac{tanbeta+tangamma}{1-tanbetatangamma}$$
we have
$$tanalpha + tanbeta + tangamma = tanalpha-tanalpha(1-tanbetatangamma) = tanalphatanbetatangamma$$
Therefore,
$$frac{text{product of radii}}{text{sum of radii}} = frac{d^3tanalphatanbetatangamma}{dtanalphatanbetatangamma} = d^2$$
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
$endgroup$
add a comment |
$begingroup$
If the angles made by the lines at their common point are $2alpha$, $2beta$, $2 gamma$ (where their sum is $2pi$), then the three radii are pretty clearly
$$d tan alpha qquad dtan beta qquad d tan gamma$$
where $d$ is the distance from the point of intersection to any of the points of tangency. (In the given problem, $d=4$.) The product of the radii is trivial; as for the sum ...
Since
$$tanalpha = tan(pi-beta-gamma) = -tan(beta+gamma) = -frac{tanbeta+tangamma}{1-tanbetatangamma}$$
we have
$$tanalpha + tanbeta + tangamma = tanalpha-tanalpha(1-tanbetatangamma) = tanalphatanbetatangamma$$
Therefore,
$$frac{text{product of radii}}{text{sum of radii}} = frac{d^3tanalphatanbetatangamma}{dtanalphatanbetatangamma} = d^2$$
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
$endgroup$
If the angles made by the lines at their common point are $2alpha$, $2beta$, $2 gamma$ (where their sum is $2pi$), then the three radii are pretty clearly
$$d tan alpha qquad dtan beta qquad d tan gamma$$
where $d$ is the distance from the point of intersection to any of the points of tangency. (In the given problem, $d=4$.) The product of the radii is trivial; as for the sum ...
Since
$$tanalpha = tan(pi-beta-gamma) = -tan(beta+gamma) = -frac{tanbeta+tangamma}{1-tanbetatangamma}$$
we have
$$tanalpha + tanbeta + tangamma = tanalpha-tanalpha(1-tanbetatangamma) = tanalphatanbetatangamma$$
Therefore,
$$frac{text{product of radii}}{text{sum of radii}} = frac{d^3tanalphatanbetatangamma}{dtanalphatanbetatangamma} = d^2$$
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
answered Sep 16 '15 at 3:26
BlueBlue
48.3k870153
48.3k870153
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$begingroup$
Suppose the lines make angles $alpha$, $beta$, $gamma$ (with $alpha + beta + gamma = 2pi$). Express the radii in terms of trig functions of $alpha/2$, etc, (and the distance $4$), and simplify.
$endgroup$
– Blue
Sep 15 '15 at 20:01
$begingroup$
@Blue I'm not being able to simplify.Please write out atleast a few steps..
$endgroup$
– user220382
Sep 15 '15 at 21:37