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632:{\displaystyle {\begin{array}{ccccc}&\displaystyle {\frac {a(b+c)^{2}}{b+c-a}}&:&\displaystyle {\frac {b(c+a)^{2}}{c+a-b}}&:&\displaystyle {\frac {c(a+b)^{2}}{a+b-c}}\\=&\sin ^{2}\!A\,\cos ^{2}{\frac {B-C}{2}}&:&\sin ^{2}\!B\,\cos ^{2}{\frac {C-A}{2}}&:&\sin ^{2}\!C\,\cos ^{2}{\frac {A-B}{2}}\end{array}}}
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The
Apollonius problem is the problem of constructing a circle tangent to three given circles in a plane. In general, there are eight circles touching three given circles. The circle
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The solution of the
Apollonius problem has been known for centuries. But the Apollonius point was first noted in 1987.
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of a triangle. This usage could also be justified on the ground that the isodynamic points are related to the three
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referred to in the above definition is one of these eight circles touching the three excircles of triangle
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Katarzyna
Wilczek (2010). "The harmonic center of a trilateral and the Apollonius point of a triangle".
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C. Kimberling; Shiko Iwata; Hidetosi
Fukagawa (1987). "Problem 1091 and Solution".
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of the three line segments joining each vertex of the triangle to the points of
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The
Apollonius point of a triangle is defined as follows.
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The trilinear coordinates of the
Apollonius point are
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198:be the circle which touches the three excircles
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693:Journal of Mathematics and Applications
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221:be the points of contact of the circle
66:" has also been used to refer to the
656:(262–190 BC), geometer and astronomer
43:(ETC). It is defined as the point of
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225:with the three excircles. The lines
233:. The point of concurrence is the
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57:is tangent to all three excircles
261:Encyclopedia of Triangle Centers
41:Encyclopedia of Triangle Centers
164:be any given triangle. Let the
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62:In the literature, the term "
74:associated with a triangle.
129: Apollonius circle of
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175:opposite to the vertices
55:and a larger circle that
51:formed by the opposing
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282:Trilinear coordinates
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743:Crux Mathematicorum
713:Kimberling, Clark.
654:Apollonius of Perga
649:Apollonius' theorem
715:"Apollonius Point"
665:Apollonian circles
660:Apollonius problem
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267:is the called the
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72:Apollonian circles
21:Euclidean geometry
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269:Apollonius circle
68:isodynamic points
64:Apollonius points
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768:Triangle centers
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717:. Archived from
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670:Isodynamic point
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147:Apollonius point
145:: concur at the
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109: Excircles
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37:Clark Kimberling
25:Apollonius point
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29:triangle center
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16:Triangle center
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721:on 10 May 2012
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95:Extended sides
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31:designated as
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227:AA', BB', CC'
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723:. Retrieved
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141: Lines
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97:of triangle
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263:the circle
45:concurrence
750:: 217–218.
678:References
231:concurrent
219:A', B', C'
81:Definition
699:: 95–101.
614:−
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561:−
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508:−
499:
455:−
398:−
341:−
166:excircles
35:(181) in
762:Category
642:See also
53:excircle
49:tangency
177:A, B, C
725:16 May
217:. Let
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127:
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23:, the
259:. In
27:is a
727:2012
229:are
157:Let
596:cos
581:sin
543:cos
528:sin
490:cos
475:sin
275:ABC
271:of
256:ABC
241:ABC
237:of
208:, E
204:, E
189:, E
185:, E
179:be
172:ABC
168:of
161:ABC
133:ABC
119:, E
115:, E
101:ABC
39:'s
19:In
764::
748:13
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705:^
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278:.
59:.
729:.
621:2
617:B
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522::
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505:B
494:2
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479:2
469:=
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431:b
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422:(
419:c
410::
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382:2
378:)
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365:(
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353::
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321:)
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311:b
308:(
305:a
273:△
265:E
254:△
250:E
244:.
239:△
223:E
215:E
210:C
206:B
202:A
200:E
196:E
191:C
187:B
183:A
181:E
170:△
159:△
131:△
121:C
117:B
113:A
111:E
99:△
33:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
