Locus of all midpoints. [closed]
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Two circles intersect at $A$ and $B$, $PAQ$ is a straight line through $A$ meeting the circles at $P$ and $Q$. Find a locus of a midpoint $PQ$.
Please give only hints and not the solution. This is a question of pure geometry and not analytic geometry.
geometry locus
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
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closed as off-topic by Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker Dec 1 '18 at 15:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Two circles intersect at $A$ and $B$, $PAQ$ is a straight line through $A$ meeting the circles at $P$ and $Q$. Find a locus of a midpoint $PQ$.
Please give only hints and not the solution. This is a question of pure geometry and not analytic geometry.
geometry locus
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
$endgroup$
closed as off-topic by Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker Dec 1 '18 at 15:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
I have a feeling that the shape might represent an ellipse but not sure how to prove.
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:04
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Do you know spiral similarity?
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– greedoid
Dec 1 '18 at 8:08
$begingroup$
No I don't know spiral similarity
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:28
1
$begingroup$
Did you even try anything?
$endgroup$
– greedoid
Dec 1 '18 at 8:52
add a comment |
$begingroup$
Two circles intersect at $A$ and $B$, $PAQ$ is a straight line through $A$ meeting the circles at $P$ and $Q$. Find a locus of a midpoint $PQ$.
Please give only hints and not the solution. This is a question of pure geometry and not analytic geometry.
geometry locus
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
$endgroup$
Two circles intersect at $A$ and $B$, $PAQ$ is a straight line through $A$ meeting the circles at $P$ and $Q$. Find a locus of a midpoint $PQ$.
Please give only hints and not the solution. This is a question of pure geometry and not analytic geometry.
geometry locus
geometry locus
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
edited Dec 1 '18 at 9:18
greedoid
38.7k114797
38.7k114797
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
asked Dec 1 '18 at 8:00
Doctorwho2311Doctorwho2311
373110
373110
closed as off-topic by Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker Dec 1 '18 at 15:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker Dec 1 '18 at 15:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, user10354138, Cesareo, Vidyanshu Mishra, gebruiker
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
I have a feeling that the shape might represent an ellipse but not sure how to prove.
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:04
$begingroup$
Do you know spiral similarity?
$endgroup$
– greedoid
Dec 1 '18 at 8:08
$begingroup$
No I don't know spiral similarity
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:28
1
$begingroup$
Did you even try anything?
$endgroup$
– greedoid
Dec 1 '18 at 8:52
add a comment |
$begingroup$
I have a feeling that the shape might represent an ellipse but not sure how to prove.
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:04
$begingroup$
Do you know spiral similarity?
$endgroup$
– greedoid
Dec 1 '18 at 8:08
$begingroup$
No I don't know spiral similarity
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:28
1
$begingroup$
Did you even try anything?
$endgroup$
– greedoid
Dec 1 '18 at 8:52
$begingroup$
I have a feeling that the shape might represent an ellipse but not sure how to prove.
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:04
$begingroup$
I have a feeling that the shape might represent an ellipse but not sure how to prove.
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:04
$begingroup$
Do you know spiral similarity?
$endgroup$
– greedoid
Dec 1 '18 at 8:08
$begingroup$
Do you know spiral similarity?
$endgroup$
– greedoid
Dec 1 '18 at 8:08
$begingroup$
No I don't know spiral similarity
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:28
$begingroup$
No I don't know spiral similarity
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:28
1
1
$begingroup$
Did you even try anything?
$endgroup$
– greedoid
Dec 1 '18 at 8:52
$begingroup$
Did you even try anything?
$endgroup$
– greedoid
Dec 1 '18 at 8:52
add a comment |
1 Answer
1
active
oldest
votes
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It is much harder to come up with a hint than answering the question.
Just stare at following picture before reading anything after it.
Let's see whether you can figure out the answer.
Let $C$ and $D$ be centers of the circles and $O = verb/mid/(C,D)$ be their midpoint.
Construct three lines perpendicular to line $PQ$, passing through $C, O, D$ respectively.
Let these three lines intersect line $PQ$ at $C', O'$ and $D'$.
Since $A$ and $P$ lies on the circle centered at $C$. $|AC| = |PC|$ and $CC'$ is the perpendicular bisector for segment $AP$. This means $C' = verb/mid/(P,A)$. By a similar argument, $D' = verb/mid/(Q,A)$.
Notice line $OO'$ is parallel to line $CC'$ and line $DD'$. Since $O = verb/mid/(C,D)$, we have
$$O' = verb/mid/(C',D') = verb/mid/(verb/mid/(P,A),verb/mid/(Q,A))
= verb/mid/(verb/mid/(P,Q),A) = verb/mid/(M,A)$$
Using the fact $OO'$ is perpendicular to $AM$, we get $|OA| = |OM|$.
From this, we can deduce the locus we seek is a circle centered at $O$ passing through $A$.
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
$endgroup$
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Fine. You should only correct a typo in the last-but-one line: $|OA|=|OM|$.
$endgroup$
– Aretino
Dec 1 '18 at 11:38
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@Aretino fixed, thanks for catching that.
$endgroup$
– achille hui
Dec 1 '18 at 11:41
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It is much harder to come up with a hint than answering the question.
Just stare at following picture before reading anything after it.
Let's see whether you can figure out the answer.
Let $C$ and $D$ be centers of the circles and $O = verb/mid/(C,D)$ be their midpoint.
Construct three lines perpendicular to line $PQ$, passing through $C, O, D$ respectively.
Let these three lines intersect line $PQ$ at $C', O'$ and $D'$.
Since $A$ and $P$ lies on the circle centered at $C$. $|AC| = |PC|$ and $CC'$ is the perpendicular bisector for segment $AP$. This means $C' = verb/mid/(P,A)$. By a similar argument, $D' = verb/mid/(Q,A)$.
Notice line $OO'$ is parallel to line $CC'$ and line $DD'$. Since $O = verb/mid/(C,D)$, we have
$$O' = verb/mid/(C',D') = verb/mid/(verb/mid/(P,A),verb/mid/(Q,A))
= verb/mid/(verb/mid/(P,Q),A) = verb/mid/(M,A)$$
Using the fact $OO'$ is perpendicular to $AM$, we get $|OA| = |OM|$.
From this, we can deduce the locus we seek is a circle centered at $O$ passing through $A$.
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
$endgroup$
$begingroup$
Fine. You should only correct a typo in the last-but-one line: $|OA|=|OM|$.
$endgroup$
– Aretino
Dec 1 '18 at 11:38
$begingroup$
@Aretino fixed, thanks for catching that.
$endgroup$
– achille hui
Dec 1 '18 at 11:41
add a comment |
$begingroup$
It is much harder to come up with a hint than answering the question.
Just stare at following picture before reading anything after it.
Let's see whether you can figure out the answer.
Let $C$ and $D$ be centers of the circles and $O = verb/mid/(C,D)$ be their midpoint.
Construct three lines perpendicular to line $PQ$, passing through $C, O, D$ respectively.
Let these three lines intersect line $PQ$ at $C', O'$ and $D'$.
Since $A$ and $P$ lies on the circle centered at $C$. $|AC| = |PC|$ and $CC'$ is the perpendicular bisector for segment $AP$. This means $C' = verb/mid/(P,A)$. By a similar argument, $D' = verb/mid/(Q,A)$.
Notice line $OO'$ is parallel to line $CC'$ and line $DD'$. Since $O = verb/mid/(C,D)$, we have
$$O' = verb/mid/(C',D') = verb/mid/(verb/mid/(P,A),verb/mid/(Q,A))
= verb/mid/(verb/mid/(P,Q),A) = verb/mid/(M,A)$$
Using the fact $OO'$ is perpendicular to $AM$, we get $|OA| = |OM|$.
From this, we can deduce the locus we seek is a circle centered at $O$ passing through $A$.
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
$endgroup$
$begingroup$
Fine. You should only correct a typo in the last-but-one line: $|OA|=|OM|$.
$endgroup$
– Aretino
Dec 1 '18 at 11:38
$begingroup$
@Aretino fixed, thanks for catching that.
$endgroup$
– achille hui
Dec 1 '18 at 11:41
add a comment |
$begingroup$
It is much harder to come up with a hint than answering the question.
Just stare at following picture before reading anything after it.
Let's see whether you can figure out the answer.
Let $C$ and $D$ be centers of the circles and $O = verb/mid/(C,D)$ be their midpoint.
Construct three lines perpendicular to line $PQ$, passing through $C, O, D$ respectively.
Let these three lines intersect line $PQ$ at $C', O'$ and $D'$.
Since $A$ and $P$ lies on the circle centered at $C$. $|AC| = |PC|$ and $CC'$ is the perpendicular bisector for segment $AP$. This means $C' = verb/mid/(P,A)$. By a similar argument, $D' = verb/mid/(Q,A)$.
Notice line $OO'$ is parallel to line $CC'$ and line $DD'$. Since $O = verb/mid/(C,D)$, we have
$$O' = verb/mid/(C',D') = verb/mid/(verb/mid/(P,A),verb/mid/(Q,A))
= verb/mid/(verb/mid/(P,Q),A) = verb/mid/(M,A)$$
Using the fact $OO'$ is perpendicular to $AM$, we get $|OA| = |OM|$.
From this, we can deduce the locus we seek is a circle centered at $O$ passing through $A$.
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
$endgroup$
It is much harder to come up with a hint than answering the question.
Just stare at following picture before reading anything after it.
Let's see whether you can figure out the answer.
Let $C$ and $D$ be centers of the circles and $O = verb/mid/(C,D)$ be their midpoint.
Construct three lines perpendicular to line $PQ$, passing through $C, O, D$ respectively.
Let these three lines intersect line $PQ$ at $C', O'$ and $D'$.
Since $A$ and $P$ lies on the circle centered at $C$. $|AC| = |PC|$ and $CC'$ is the perpendicular bisector for segment $AP$. This means $C' = verb/mid/(P,A)$. By a similar argument, $D' = verb/mid/(Q,A)$.
Notice line $OO'$ is parallel to line $CC'$ and line $DD'$. Since $O = verb/mid/(C,D)$, we have
$$O' = verb/mid/(C',D') = verb/mid/(verb/mid/(P,A),verb/mid/(Q,A))
= verb/mid/(verb/mid/(P,Q),A) = verb/mid/(M,A)$$
Using the fact $OO'$ is perpendicular to $AM$, we get $|OA| = |OM|$.
From this, we can deduce the locus we seek is a circle centered at $O$ passing through $A$.
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
edited Dec 1 '18 at 11:41
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
answered Dec 1 '18 at 11:29
achille huiachille hui
95.6k5131258
95.6k5131258
$begingroup$
Fine. You should only correct a typo in the last-but-one line: $|OA|=|OM|$.
$endgroup$
– Aretino
Dec 1 '18 at 11:38
$begingroup$
@Aretino fixed, thanks for catching that.
$endgroup$
– achille hui
Dec 1 '18 at 11:41
add a comment |
$begingroup$
Fine. You should only correct a typo in the last-but-one line: $|OA|=|OM|$.
$endgroup$
– Aretino
Dec 1 '18 at 11:38
$begingroup$
@Aretino fixed, thanks for catching that.
$endgroup$
– achille hui
Dec 1 '18 at 11:41
$begingroup$
Fine. You should only correct a typo in the last-but-one line: $|OA|=|OM|$.
$endgroup$
– Aretino
Dec 1 '18 at 11:38
$begingroup$
Fine. You should only correct a typo in the last-but-one line: $|OA|=|OM|$.
$endgroup$
– Aretino
Dec 1 '18 at 11:38
$begingroup$
@Aretino fixed, thanks for catching that.
$endgroup$
– achille hui
Dec 1 '18 at 11:41
$begingroup$
@Aretino fixed, thanks for catching that.
$endgroup$
– achille hui
Dec 1 '18 at 11:41
add a comment |
$begingroup$
I have a feeling that the shape might represent an ellipse but not sure how to prove.
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:04
$begingroup$
Do you know spiral similarity?
$endgroup$
– greedoid
Dec 1 '18 at 8:08
$begingroup$
No I don't know spiral similarity
$endgroup$
– Doctorwho2311
Dec 1 '18 at 8:28
1
$begingroup$
Did you even try anything?
$endgroup$
– greedoid
Dec 1 '18 at 8:52