Power of a point: Minimize the length of a chord of a circle containing A and B
$begingroup$
Let A, B be points lying on distinct sides of a line k. Find a
circle o through A, B such that the length of the chord of o lying inside k is
minimal.
Intuitively, it seems to me that the circle that will minimize the chord is the circle with points A and B lying on the same diameter (i.e. the segment AB is a diameter of the circle). I just do not know where to start or how to rigorously prove such a circle minimizes the chord.
Any help would be much appreciated.
geometry euclidean-geometry
asked Dec 7 '18 at 8:19
ricorico
706
$endgroup$
add a comment |
$begingroup$
Let A, B be points lying on distinct sides of a line k. Find a
circle o through A, B such that the length of the chord of o lying inside k is
minimal.
Intuitively, it seems to me that the circle that will minimize the chord is the circle with points A and B lying on the same diameter (i.e. the segment AB is a diameter of the circle). I just do not know where to start or how to rigorously prove such a circle minimizes the chord.
Any help would be much appreciated.
geometry euclidean-geometry
asked Dec 7 '18 at 8:19
ricorico
706
$endgroup$
$begingroup$
What do you mean by "lying on distinct sides"?
$endgroup$
– RcnSc
Dec 7 '18 at 8:40
$begingroup$
Nice question. @RcnSc, he means opposite sides.
$endgroup$
– Anubhab Ghosal
Dec 7 '18 at 9:02
add a comment |
$begingroup$
Let A, B be points lying on distinct sides of a line k. Find a
circle o through A, B such that the length of the chord of o lying inside k is
minimal.
Intuitively, it seems to me that the circle that will minimize the chord is the circle with points A and B lying on the same diameter (i.e. the segment AB is a diameter of the circle). I just do not know where to start or how to rigorously prove such a circle minimizes the chord.
Any help would be much appreciated.
geometry euclidean-geometry
asked Dec 7 '18 at 8:19
ricorico
706
$endgroup$
Let A, B be points lying on distinct sides of a line k. Find a
circle o through A, B such that the length of the chord of o lying inside k is
minimal.
Intuitively, it seems to me that the circle that will minimize the chord is the circle with points A and B lying on the same diameter (i.e. the segment AB is a diameter of the circle). I just do not know where to start or how to rigorously prove such a circle minimizes the chord.
Any help would be much appreciated.
geometry euclidean-geometry
geometry euclidean-geometry
asked Dec 7 '18 at 8:19
ricorico
706
asked Dec 7 '18 at 8:19
ricorico
706
asked Dec 7 '18 at 8:19
ricorico
706
asked Dec 7 '18 at 8:19
ricorico
706
asked Dec 7 '18 at 8:19
ricorico
706
706
$begingroup$
What do you mean by "lying on distinct sides"?
$endgroup$
– RcnSc
Dec 7 '18 at 8:40
$begingroup$
Nice question. @RcnSc, he means opposite sides.
$endgroup$
– Anubhab Ghosal
Dec 7 '18 at 9:02
add a comment |
$begingroup$
What do you mean by "lying on distinct sides"?
$endgroup$
– RcnSc
Dec 7 '18 at 8:40
$begingroup$
Nice question. @RcnSc, he means opposite sides.
$endgroup$
– Anubhab Ghosal
Dec 7 '18 at 9:02
$begingroup$
What do you mean by "lying on distinct sides"?
$endgroup$
– RcnSc
Dec 7 '18 at 8:40
$begingroup$
What do you mean by "lying on distinct sides"?
$endgroup$
– RcnSc
Dec 7 '18 at 8:40
$begingroup$
Nice question. @RcnSc, he means opposite sides.
$endgroup$
– Anubhab Ghosal
Dec 7 '18 at 9:02
$begingroup$
Nice question. @RcnSc, he means opposite sides.
$endgroup$
– Anubhab Ghosal
Dec 7 '18 at 9:02
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Suppose the line $k$ and $AB$ meet at $E$. Let $C$ and $D$ be the points of intersection of $k$ with the variable circle through $A$ and $B$. We wish to minimize $CD$.
$CD=CE+EDgeq 2sqrt{CE.ED}=2sqrt{AE.EB}$. Equality is attained when $CE=ED$, that is when the center of the circle $O$ lies on the line perpendicular to $k$ through $E$( and $O$ must also lie on the perpendicular bisector of $AB$.)
$blacksquare$
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
$endgroup$
add a comment |
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1 Answer
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$begingroup$
Suppose the line $k$ and $AB$ meet at $E$. Let $C$ and $D$ be the points of intersection of $k$ with the variable circle through $A$ and $B$. We wish to minimize $CD$.
$CD=CE+EDgeq 2sqrt{CE.ED}=2sqrt{AE.EB}$. Equality is attained when $CE=ED$, that is when the center of the circle $O$ lies on the line perpendicular to $k$ through $E$( and $O$ must also lie on the perpendicular bisector of $AB$.)
$blacksquare$
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
$endgroup$
add a comment |
$begingroup$
Suppose the line $k$ and $AB$ meet at $E$. Let $C$ and $D$ be the points of intersection of $k$ with the variable circle through $A$ and $B$. We wish to minimize $CD$.
$CD=CE+EDgeq 2sqrt{CE.ED}=2sqrt{AE.EB}$. Equality is attained when $CE=ED$, that is when the center of the circle $O$ lies on the line perpendicular to $k$ through $E$( and $O$ must also lie on the perpendicular bisector of $AB$.)
$blacksquare$
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
$endgroup$
add a comment |
$begingroup$
Suppose the line $k$ and $AB$ meet at $E$. Let $C$ and $D$ be the points of intersection of $k$ with the variable circle through $A$ and $B$. We wish to minimize $CD$.
$CD=CE+EDgeq 2sqrt{CE.ED}=2sqrt{AE.EB}$. Equality is attained when $CE=ED$, that is when the center of the circle $O$ lies on the line perpendicular to $k$ through $E$( and $O$ must also lie on the perpendicular bisector of $AB$.)
$blacksquare$
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
$endgroup$
Suppose the line $k$ and $AB$ meet at $E$. Let $C$ and $D$ be the points of intersection of $k$ with the variable circle through $A$ and $B$. We wish to minimize $CD$.
$CD=CE+EDgeq 2sqrt{CE.ED}=2sqrt{AE.EB}$. Equality is attained when $CE=ED$, that is when the center of the circle $O$ lies on the line perpendicular to $k$ through $E$( and $O$ must also lie on the perpendicular bisector of $AB$.)
$blacksquare$
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
answered Dec 7 '18 at 9:11
Anubhab GhosalAnubhab Ghosal
1,20119
1,20119
add a comment |
add a comment |
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$begingroup$
What do you mean by "lying on distinct sides"?
$endgroup$
– RcnSc
Dec 7 '18 at 8:40
$begingroup$
Nice question. @RcnSc, he means opposite sides.
$endgroup$
– Anubhab Ghosal
Dec 7 '18 at 9:02