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radical axis
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Midpoint lies on Radical axis
Ankoganit   2
N Jul 1, 2016 by Ghost_rider
Source: My friend WizardMath
$C_1$ and $C_2$ are two disjoint circles. $AB$ is an external common tangent and $CD$ is an internal common tangent to them, such that $C$ is nearer to $AB$ than $D$. Suppose $P$ is the pole of $BC$ wrt $C_2$ and $M$ is the intersection point of the polars of $P$ wrt $C_1$ and $C_2$. Prove that the midpoint of $PM$ lies on the radical axis of $C_1$ and $C_2$.
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Ankoganit
May 6, 2016
Ghost_rider
Jul 1, 2016
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Midpoint lies on Radical axis
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geometry
radical axis
power of a point
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Source: My friend WizardMath
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Ankoganit
3070 posts
Ankoganit
#1 h  May 6, 2016, 3:09 PM • 3 Y
Y by sarthak7, Adventure10, Mango247
$C_1$ and $C_2$ are two disjoint circles. $AB$ is an external common tangent and $CD$ is an internal common tangent to them, such that $C$ is nearer to $AB$ than $D$. Suppose $P$ is the pole of $BC$ wrt $C_2$ and $M$ is the intersection point of the polars of $P$ wrt $C_1$ and $C_2$. Prove that the midpoint of $PM$ lies on the radical axis of $C_1$ and $C_2$.
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Ankoganit
3070 posts
Ankoganit
#2 h  May 25, 2016, 2:18 PM • 1 Y
Y by Adventure10
Any solutions? :roll:
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Ghost_rider
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Ghost_rider
#3 h  Jul 1, 2016, 2:48 AM • 2 Y
Y by Adventure10, Mango247
Let $DC$ $\cap$ $AB$ = $P$, $CB$ $\cap$ $AD$ = $M$ and midpoint of $PM$ = $R$. It´s easy to see that $P$ and $M$ are conjugate points respect to $C_1$ and $C_2$.Then the circle of diameter $MP$ is ortogonal to $C_1$ and $C_2$ and his radio is the tangent of $C_1$ and $C_2$ .Then $R$ lies on the radical axis of $C_1$ and $C_2$.
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