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gives the coordinates of the point. It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides. The
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If ABC is a right triangle with right angle at A, this statement needs to be modified by dropping the reference to AA' since the point A' does not exist.
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Beban-Brkić, J.; Volenec, V.; Kolar-Begović, Z.; Kolar-Šuper, R. (2013), "On
Gergonne point of the triangle in isotropic plane",
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An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the
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the symmedian point appears as the sixth point, X(6). For a non-equilateral triangle, it lies in the open
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53:(red). The symmedians intersect in the symmedian point L, the angle bisectors in the
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of a triangle is the same as the symmedian point of the triangle's
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called its existence "one of the crown jewels of modern geometry".
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is a symmedian, as can be seen by drawing the circle with center
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Episodes in
Nineteenth and Twentieth Century Euclidean Geometry
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punctured at its own center, and could be any point therein.
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proved the existence of the symmedian point in 1873, and
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Intersection of the three symmedian lines of a triangle
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The symmedian point of a triangle with side lengths
253:For the extension to an irregular tetrahedron see
334:Bradley, Christopher J.; Smith, Geoff C. (2006),
159:can be constructed in the following way: let the
136:of the corresponding lines. The solution of this
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360:Rad Hrvatske Akademije Znanosti i Umjetnosti
300:(1995), "Chapter 7: The Symmedian Point",
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211:intersect at the symmedian point of
336:"The locations of triangle centers"
306:Mathematical Association of America
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250:had also noted the point in 1809.
155:The symmedian point of a triangle
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246:published a paper on it in 1847.
81:is the intersection of the three
322:Encyclopedia of Triangle Centers
104:Encyclopedia of Triangle Centers
32:Lemoine Point Conservation Area
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248:Simon Antoine Jean L'Huilier
89:reflected at the associated
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238:The French mathematician
179:, and analogously define
57:I and the medians in the
30:Not to be confused with
163:of the circumcircle of
324:, accessed 2014-11-06.
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138:overdetermined system
127:trilinear coordinates
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304:, Washington, D.C.:
142:least squares method
108:orthocentroidal disk
340:Forum Geometricorum
244:Ernst Wilhelm Grebe
217:Brianchon's theorem
193:tangential triangle
391:Weisstein, Eric W.
134:hesse normal forms
93:) of a triangle.
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394:"Symmedian Point"
16:(Redirected from
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79:symmedian point
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18:Symmedian point
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366:: 95–106,
281:References
83:symmedians
51:symmedians
399:MathWorld
255:symmedian
45:(black),
412:Category
227:through
175:meet at
167:through
67:geometry
59:centroid
55:incenter
372:3100227
346:: 57–70
219:. Line
191:is the
187:; then
102:In the
87:medians
43:medians
370:
189:A'B'C'
69:, the
261:Notes
231:and
207:and
183:and
171:and
121:and
221:AA'
213:ABC
209:CC'
205:BB'
201:AA'
197:ABC
195:of
165:ABC
157:ABC
77:or
65:In
414::
396:.
368:MR
364:17
362:,
342:,
338:,
314:^
289:^
257:.
235:.
225:A'
203:,
185:C'
181:B'
177:A'
152:.
129:.
117:,
73:,
61:G.
402:.
375:.
348:.
344:6
309:.
233:C
229:B
173:C
169:B
123:c
119:b
115:a
85:(
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