Apollonius special case.
$begingroup$
Given circles $c$ and $d$ (green circles) and chord $AB$ arranged as shown in the figure below. How can I construct the blue circle tangent to all previous elements?
[Once I think I found an elementary construction which unfortunately I cannot remember.]
geometry euclidean-geometry geometric-transformation
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
$endgroup$
add a comment |
$begingroup$
Given circles $c$ and $d$ (green circles) and chord $AB$ arranged as shown in the figure below. How can I construct the blue circle tangent to all previous elements?
[Once I think I found an elementary construction which unfortunately I cannot remember.]
geometry euclidean-geometry geometric-transformation
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
$endgroup$
add a comment |
$begingroup$
Given circles $c$ and $d$ (green circles) and chord $AB$ arranged as shown in the figure below. How can I construct the blue circle tangent to all previous elements?
[Once I think I found an elementary construction which unfortunately I cannot remember.]
geometry euclidean-geometry geometric-transformation
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
$endgroup$
Given circles $c$ and $d$ (green circles) and chord $AB$ arranged as shown in the figure below. How can I construct the blue circle tangent to all previous elements?
[Once I think I found an elementary construction which unfortunately I cannot remember.]
geometry euclidean-geometry geometric-transformation
geometry euclidean-geometry geometric-transformation
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
edited Dec 20 '18 at 15:06
greedoid
40.5k1148100
40.5k1148100
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
asked Dec 20 '18 at 13:09
nickchalkidanickchalkida
930817
930817
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: Let small green circle touch big one and a chord at $M$ and $N$. Then it is not difficult to see that line $MN$ cuts small arc $AB$ at it midpoint, say $S$ (check it by, say homothety). From $S$ draw a tangent to small green and let it cut $AB$ at $P$ and touch small green at $D$. Now draw a angle bisector for angle $angle BPD$. Blue circle has center on it and touch a line $SD$ at $D$. (All these can be confirmed by radical axsis and/or inversion at $S$.)
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
$endgroup$
$begingroup$
Great answer! Thanks a lot. That what I was looking for!
$endgroup$
– nickchalkida
Dec 20 '18 at 13:37
$begingroup$
Will you accept an answer since you like it?
$endgroup$
– greedoid
Dec 26 '18 at 20:58
add a comment |
Your Answer
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Hint: Let small green circle touch big one and a chord at $M$ and $N$. Then it is not difficult to see that line $MN$ cuts small arc $AB$ at it midpoint, say $S$ (check it by, say homothety). From $S$ draw a tangent to small green and let it cut $AB$ at $P$ and touch small green at $D$. Now draw a angle bisector for angle $angle BPD$. Blue circle has center on it and touch a line $SD$ at $D$. (All these can be confirmed by radical axsis and/or inversion at $S$.)
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
$endgroup$
$begingroup$
Great answer! Thanks a lot. That what I was looking for!
$endgroup$
– nickchalkida
Dec 20 '18 at 13:37
$begingroup$
Will you accept an answer since you like it?
$endgroup$
– greedoid
Dec 26 '18 at 20:58
add a comment |
$begingroup$
Hint: Let small green circle touch big one and a chord at $M$ and $N$. Then it is not difficult to see that line $MN$ cuts small arc $AB$ at it midpoint, say $S$ (check it by, say homothety). From $S$ draw a tangent to small green and let it cut $AB$ at $P$ and touch small green at $D$. Now draw a angle bisector for angle $angle BPD$. Blue circle has center on it and touch a line $SD$ at $D$. (All these can be confirmed by radical axsis and/or inversion at $S$.)
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
$endgroup$
$begingroup$
Great answer! Thanks a lot. That what I was looking for!
$endgroup$
– nickchalkida
Dec 20 '18 at 13:37
$begingroup$
Will you accept an answer since you like it?
$endgroup$
– greedoid
Dec 26 '18 at 20:58
add a comment |
$begingroup$
Hint: Let small green circle touch big one and a chord at $M$ and $N$. Then it is not difficult to see that line $MN$ cuts small arc $AB$ at it midpoint, say $S$ (check it by, say homothety). From $S$ draw a tangent to small green and let it cut $AB$ at $P$ and touch small green at $D$. Now draw a angle bisector for angle $angle BPD$. Blue circle has center on it and touch a line $SD$ at $D$. (All these can be confirmed by radical axsis and/or inversion at $S$.)
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
$endgroup$
Hint: Let small green circle touch big one and a chord at $M$ and $N$. Then it is not difficult to see that line $MN$ cuts small arc $AB$ at it midpoint, say $S$ (check it by, say homothety). From $S$ draw a tangent to small green and let it cut $AB$ at $P$ and touch small green at $D$. Now draw a angle bisector for angle $angle BPD$. Blue circle has center on it and touch a line $SD$ at $D$. (All these can be confirmed by radical axsis and/or inversion at $S$.)
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
answered Dec 20 '18 at 13:26
greedoidgreedoid
40.5k1148100
40.5k1148100
$begingroup$
Great answer! Thanks a lot. That what I was looking for!
$endgroup$
– nickchalkida
Dec 20 '18 at 13:37
$begingroup$
Will you accept an answer since you like it?
$endgroup$
– greedoid
Dec 26 '18 at 20:58
add a comment |
$begingroup$
Great answer! Thanks a lot. That what I was looking for!
$endgroup$
– nickchalkida
Dec 20 '18 at 13:37
$begingroup$
Will you accept an answer since you like it?
$endgroup$
– greedoid
Dec 26 '18 at 20:58
$begingroup$
Great answer! Thanks a lot. That what I was looking for!
$endgroup$
– nickchalkida
Dec 20 '18 at 13:37
$begingroup$
Great answer! Thanks a lot. That what I was looking for!
$endgroup$
– nickchalkida
Dec 20 '18 at 13:37
$begingroup$
Will you accept an answer since you like it?
$endgroup$
– greedoid
Dec 26 '18 at 20:58
$begingroup$
Will you accept an answer since you like it?
$endgroup$
– greedoid
Dec 26 '18 at 20:58
add a comment |
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