Orthic Triangle

Author

Eric W. Weisstein
April 2, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/OrthicTriangle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/OrthicTriangle.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Figure

t=Triangle[{{0.,0.},{3.,0.},{1.3,2.2}}];
Show[Graphics[{
t,
{Red,ht=OrthicTriangle[t]},
{
Blue,
Altitudes[t],
PerpSquare[#,.1,3]&/@Transpose[{Altitudes[t],Sides[t]}]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Point/@Vertices[ht],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
{{Orthocenter[t],"H",Red}}
],
{PointSize[.025],LabelOffset->.08}
]
}],AspectRatio->Automatic]

Vertices

<<Utilities`Typesetting`

CevianTriangle[Orthocenter]//MatrixForm//TraditionalForm

Side Lengths

FullSimplify[DistanceTrilinears[SideLengths[CevianTriangle[Orthocenter]]],TriangleQ[]]

FullSimplify[Abs[%],TriangleQ[]]

abc=Simplify[LawOfCosines[%],TriangleQ[]]

Acute

FullSimplify[abc,AcuteQ[]]

Semiperimeter

Factor[FullSimplify[Plus@@DistanceTrilinears[abc]/2,TriangleQ[]]]

sh=FullSimplify[LawOfCosines[%],TriangleQ[]]

LawOfCosines[Factor[Together//@DistanceTrilinears[Refine[sh,Pi/2<=A<Pi&&0<B<Pi/2&&0<C<Pi/2]]]]

LawOfCosines[Factor[Together//@DistanceTrilinears[Refine[sh,Pi/2<=B<Pi&&0<A<Pi/2&&0<C<Pi/2]]]]

LawOfCosines[Factor[Together//@DistanceTrilinears[Refine[sh,Pi/2<=C<Pi&&0<A<Pi/2&&0<B<Pi/2]]]]

Acute

FullSimplify[DistanceTrilinears[%]Δ/Area[] Circumradius[]/R,AcuteQ[]]

Circumradius

FullSimplify[LawOfCosines[FullSimplify[Together//@DistanceTrilinears[Simplify[Circumradius[abc]R/Circumradius[],TriangleQ[]&&Pi/2<A<Pi&&B<Pi/2&&C<Pi/2]],TriangleQ[]]],TriangleQ[]&&Pi/2<A<Pi&&B<Pi/2&&C<Pi/2]

Acute

FullSimplify[LawOfCosines[Together//@DistanceTrilinears[Simplify[Circumradius[abc],AcuteQ[]]]R/Circumradius[]],TriangleQ[]&&AcuteQ[]]

Inradius

rh=FullSimplify[LawOfCosines[Together//@DistanceTrilinears[Simplify[Inradius[abc],TriangleQ[]]]R/Circumradius[]],TriangleQ[]]

Area

Timing[area=FullSimplify[DistanceTrilinears[CevianTriangleArea[Orthocenter]],TriangleQ[]]]

FullSimplify[LawOfCosines[area] Circumradius[]/R,TriangleQ[]]

Acute

FullSimplify[%,AcuteQ[]]

Points

Centroid

Centroid[OrthicTriangle]

g=Symmetrize[-ToAngles[Symmetrize[Factor[ToSidelengths[Centroid[OrthicTriangle]]]]]]

IncidentCenters[KimberlingCenters,g]//Timing

Circumcenter

Circumcenter[OrthicTriangle]

o=Symmetrize[-ToAngles[Symmetrize[ToSidelengths[Circumcenter[OrthicTriangle]]]]]

IncidentCenters[KimberlingCenters,o]//Timing

Name[X[5]]

Incenter

Incenter[OrthicTriangle]

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Incenter[OrthicTriangle[t]]

Orthocenter[t]

orthocenter

CenterFunction[Orthocenter]

Orthocenter

h=Symmetrize[ToAngles[Symmetrize[ToSidelengths[Orthocenter[OrthicTriangle]]]]]

IncidentCenters[KimberlingCenters,h]//Timing

Symmedian Point

...a work in progress...

t=OrthicTriangle

Show[Graphics[{
t,
{Blue,OrthicTriangle[t]},
Medians[OrthicTriangle[t]],
PointLabels[Join[
{
{Cartesian[k[t]],"K",Red},
{Cartesian[g[t]],"G",Blue}
}
]]
}]]

k=Intersections@@Take[(klines=Line@@@Transpose[{Vertices[Triangle[OrthicTriangle]],Reflection[g,#]&/@AngleBisectors[OrthicTriangle]}]),2];

t=Triangle[{{0,0},{1,0},{.3,.7}}];

SymmedianPoint[OrthicTriangle[t]]

Cartesian[k[t]]

Cartesian[CyclicTrilinears[Tan[A]Cos[B-C]][t]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[Tan[A]Cos[B-C]]]//Timing

Symmedian Figure

tri={{0.,0.},{1.3,2.2},{3.,0.}};
o=OrthicTriangle[tri][[1]];
alt=AltitudeFeet[tri];
ktri=Table[Join[{tri[[i]]},o[[{addn[i,1,3],addn[i,2,3]}]]],{i,3}];
k=SymmedianPoint/@ktri;
m=Midpoints[tri];

pts={
	tri[[1]],"(Subscript[A, 1])",tri[[2]],"(Subscript[A, 2])",tri[[3]],"(Subscript[A, 3])",
alt[[1]],"(Subscript[H, 1])",alt[[2]],"(Subscript[H, 2])",alt[[3]],"(Subscript[H, 3])",
				Orthocenter,"H",Centroid,"G",
		k[[1]],"(Subscript[K, 1])",k[[2]],"(Subscript[K, 2])",k[[3]],"(Subscript[K, 3])",
			m[[1]],"(Subscript[M, 3])",m[[2]],"(Subscript[M, 1])",m[[3]],"(Subscript[M, 2])"
};
Show[Graphics[{
	Triangle[tri],
	{Dashing[{.02,.02}],Altitudes[tri]},
			Table[Line[{tri[[i]],m[[addn[i,1,3]]]} /. Point->Identity],{i,3}],
	PointLabels[tri,pts],
	{Thickness[.01],OrthicTriangle[tri]}
}],AspectRatio->Automatic]

Euler Lines

t=Triangle[{{0.,0.},{1.2,1.9},{3.,0.}}];
h=OrthicTriangle[t];
size=PointSize[.015];
corner=Triangle/@(Flatten[#,1]&/@Transpose[{List/@Vertices[t],RotateLeft[Partition[Vertices[h],2,1,1]]}]);
euler=EulerLine[#,5,5]&/@corner;
Show[Graphics[{
{Blue,euler},
{Orange,Altitudes[t]},
t,
PointLabels[Transpose[{Point/@Vertices[t],{"A","B","C"}}],{size}],
{DarkGreen,NinePointCircle[t],
PointLabels[Transpose[{Midpoints[t],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],{size}]
},
{Red,h,
PointLabels[Transpose[{Flatten[{Orthocenter[t],Point/@Vertices[h]}],{"H","(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],{size}]
},
PointLabels[Transpose[{EulerPoints[t],{"(Subscript[E, A])","(Subscript[E, B])","(Subscript[E, C])"}}],{Orange,size}],
{PointSize[.015],Blue,Intersections@@Take[euler,2]}
},AspectRatio->Automatic]]

Sides Parallel to Circumcircle tangents

t=Triangle[Reverse/@{{0.,0.},{1.2,2.3},{3.,0.}}];
v=Point/@Vertices[t];
h=OrthicTriangle[t];
Show[Graphics[{
t,
{Red,
h,
Dashing[{.02}],Altitudes[t],
PointLabels[Transpose[{Point/@Vertices[OrthicTriangle[t]],{"(Subscript[H, 1])","(Subscript[H, 2])","(Subscript[H, 3])"}}]]},
{Blue,
TangentialTriangle[t],
PointLabels[Transpose[{Point/@Vertices[TangentialTriangle[t]],{"(Subscript[T, 1])","(Subscript[T, 2])","(Subscript[T, 3])"}}]]
},
{
LimeGreen,
Circumcircle[t],
Line[{Coordinates[o],#}]&/@Vertices[t]
},
PointLabels[{{v[[1]],"(Subscript[A, 1])"},{v[[2]],"(Subscript[A, 2])"},{v[[3]],"(Subscript[A, 3])"},
{o=Circumcenter[t],"O",LimeGreen}}]
}],AspectRatio->Automatic]

Centers

(pts={#,KimberlingLookup[#[OrthicTriangle]]}&/@Take[KimberlingCenters,All]);//Timing

l=Map[First,DeleteCases[pts,_?(Last[#]==={}&)],{2}]

moses={{1,4},{2,51},{3,5},{4,52},{5,143},{6,53},{7,1843},{8,185},{9,6},{10,389},{11,1112},{19,571},{30,1154},{35,1594},{36,403},{40,3},{43,2052},{44,1990},{46,24},{48,1879},{55,427},{56,235},{57,25},{63,184},{71,570},{74,128},{80,1986},{84,155},{90,2904},{98,129},{99,130},{100,125},{101,115},{102,131},{103,114},{104,113},{105,132},{106,133},{107,134},{108,135},{109,136},{110,137},{111,138},{112,139},{164,1},{165,2},{166,7},{167,8},{168,9},{169,32},{170,76},{173,19},{174,1824},{177,1829},{188,65},{191,54},{200,1899},{236,2262},{241,460},{258,33},{259,1826},{260,429},{266,225},{289,1856},{354,428},{361,158},{362,31},{363,57},{399,1263},{442,973},{484,186},{503,92},{504,48},{505,46},{513,523},{514,512},{516,511},{517,30},{518,1503},{522,924},{523,1510},{527,2393},{528,2781},{573,216},{610,2165},{644,1562},{649,647},{650,2501},{657,2489},{672,232},{844,75},{845,63},{846,275},{900,526},{910,230},{926,804},{927,2679},{978,1093},{999,1596},{1054,107},{1128,1844},{1130,1838},{1145,974},{1155,468},{1158,1147},{1276,16},{1277,15},{1282,98},{1292,127},{1293,122},{1381,1312},{1382,1313},{1385,546},{1445,1974},{1488,1827},{1490,68},{1507,1136},{1635,1637},{1697,1593},{1709,1993},{1721,317},{1742,264},{1743,393},{1745,847},{1754,467},{1758,421},{1763,2351},{1764,418},{1766,577},{1768,110},{2077,2072},{2082,1968},{2089,1828},{2136,64},{2173,231},{2183,3003},{2265,3018},{2270,1609},{2291,1560},{2448,1113},{2449,1114},{2801,542},{2802,2777},{2808,2782},{2809,2794},{2810,2790},{2820,2799},{2821,2797},{2832,2848},{2938,95},{2939,96},{2940,252},{2941,97},{2942,1799},{2943,1105},{2948,1141},{2951,69},{2956,254},{2957,250},{2958,249}};

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm

X[1]

KimberlingLookup[X[1][OrthicTriangle]]

expr=-ToAngles[Symmetrize[Simplify[Together//@ToSidelengths[X[1][OrthicTriangle]]/.Sqrt[s_]:>Simplify[Sqrt[s],TriangleQ[]]]]]

KimberlingLookup[%]

Symmetrize[ToAngles[Simplify[expr,AcuteQ[]&&TriangleQ[]]]]

KimberlingLookup[%]

Acute-Only

KimberlingLookup[X[7][OrthicTriangle]]

AcuteQ[]

Simplify[Together//@ToSidelengths[X[7][OrthicTriangle]]/.Sqrt[s_]:>Simplify[Sqrt[s],TriangleQ[]&&AcuteQ[]],TriangleQ[]&&AcuteQ[]]

Symmetrize

Factor[%49(((((-(a^2)+(b^2)+(c^2)))^3) ((((a^2)+(b^2)-(c^2)))^3) ((((a^2)-(b^2)+(c^2)))^3))/(512 (a^8) (b^8) (c^8)))]