# 三角形の心

## 定義

ユークリッド平面上に三角形 $$ABC$$ を固定する。平面上の点 $$X$$ が実数の三つ組 $$\lambda,\mu,\nu$$ を使って位置ベクトルに関して $$(\lambda+\mu+\nu)X = \lambda A + \mu B + \nu C$$ と表せるとき、$$X=\lambda:\mu:\nu$$ と書く。

• 常に $$f(a,b,c)=f(a,c,b)$$ が成り立つ。
• ある定数 $$n$$ が存在し、任意の正実数 $$k$$ に対して常に $$f(ka,kb,kc)=k^nf(a,b,c)$$ が成り立つ。

## 例

X1

incenter

$$f(a,b,c)=a$$

X2

centroid

$$f(a,b,c)=1$$

X3

circumcenter

$$f(a,b,c)=a^2(b^2+c^2-a^2)$$

X4

orthocenter

$$f(a,b,c)=(c^2+a^2-b^2)(a^2+b^2-c^2)$$

X5

nine-point center (midpoint of the circumcenter and the orthocenter)

$$f(a,b,c)=a^2(b^2+c^2)-(b^2-c^2)^2$$

X6

symmedian point (Lemoine point, Grebe point)

$$f(a,b,c)=a^2$$

X7
ジェルゴンヌ点
Gergonne point

$$f(a,b,c)=\dfrac{1}{b+c-a}$$

X8
ナーゲル点
Nagel point

$$f(a,b,c)=b+c-a$$

X9
ミッテンプンクト
Mittenpunkt

$$f(a,b,c)=a(b+c-a)$$

X10
シュピーカー心（内心とナーゲル点の中点）
Spieker center (midpoint of the incenter and the Nagel point)

$$f(a,b,c)=b+c$$

X11
フォイエルバッハ点
Feuerbach point

$$f(a,b,c)=(b+c-a)(b-c)^2$$

X12

harmonic conjugate of the Feuerbach point with respect to the incenter and the nine-point center

$$f(a,b,c)=\dfrac{(b+c)^2}{b+c-a}$$

X13

first isogonic center (first Fermat point, Fermat point, Torricelli point)

$$f(a,b,c)=[\sqrt{3}(c^2+a^2-b^2)+4\Delta][\sqrt{3}(a^2+b^2-c^2)+4\Delta]$$   ただし、$$\Delta$$は元の三角形の面積

$$f_1(a,b,c)=a^4+a^2(b^2+c^2+4\sqrt{3}\Delta)-2(b^2-c^2)^2$$   ただし、$$\Delta$$は元の三角形の面積

X14

second isogonic center (second Fermat point)

$$f(a,b,c)=[\sqrt{3}(c^2+a^2-b^2)-4\Delta][\sqrt{3}(a^2+b^2-c^2)-4\Delta]$$   ただし、$$\Delta$$は元の三角形の面積

$$f_1(a,b,c)=a^4+a^2(b^2+c^2-4\sqrt{3}\Delta)-2(b^2-c^2)^2$$   ただし、$$\Delta$$は元の三角形の面積

X15

first isodynamic point

$$f(a,b,c)=a^2[\sqrt{3}(b^2+c^2-a^2)+4\Delta]$$   ただし、$$\Delta$$は元の三角形の面積

X16

second isodynamic point

$$f(a,b,c)=a^2[\sqrt{3}(b^2+c^2-a^2)-4\Delta]$$   ただし、$$\Delta$$は元の三角形の面積

X17

first Napoleon point

$$f(a,b,c)=(c^2+a^2-b^2+4\sqrt{3}\Delta)(a^2+b^2-c^2+4\sqrt{3}\Delta)$$   ただし、$$\Delta$$は元の三角形の面積

$$f_1(a,b,c)=-a^4+a^2(3(b^2+c^2)+4\sqrt{3}\Delta)-2(b^2-c^2)^2$$   ただし、$$\Delta$$は元の三角形の面積

X18

second Napoleon point

$$f(a,b,c)=(c^2+a^2-b^2-4\sqrt{3}\Delta)(a^2+b^2-c^2-4\sqrt{3}\Delta)$$   ただし、$$\Delta$$は元の三角形の面積

$$f_1(a,b,c)=-a^4+a^2(3(b^2+c^2)-4\sqrt{3}\Delta)-2(b^2-c^2)^2$$   ただし、$$\Delta$$は元の三角形の面積

X19
クローソン点
Clawson point

$$f(a,b,c)=a(c^2+a^2-b^2)(a^2+b^2-c^2)$$

X20
ド・ロンシャン点
de Longchamps point

$$f(a,b,c)=-3a^4+2a^2(b^2+c^2)+(b^2-c^2)^2$$

X21
シフラー点
Schiffler point

$$f(a,b,c)=\dfrac{a(b+c-a)}{b+c}$$

X22
エクセター点
Exeter point

$$f(a,b,c)=a^2(b^4+c^4-a^4)$$

X23

far-out point

$$f(a,b,c)=a^2(b^4+c^4-a^4-b^2c^2)$$

X24

perspector of the reference and orthic-of-orthic triangles

$$f(a,b,c)=a^2[2b^2c^2-(b^2+c^2-a^2)^2](c^2+a^2-b^2)(a^2+b^2-c^2)$$

X25

homothetic center of the orthic and tangential triangles

$$f(a,b,c)=a^2(c^2+a^2-b^2)(a^2+b^2-c^2)$$

X26

circumcenter of the tangential triangle

$$f(a,b,c)=a^2[b^4(c^2+a^2-b^2)^2+c^4(a^2+b^2-c^2)^2-a^4(b^2+c^2-a^2)^2+2a^2b^2c^2(b^2+c^2-a^2))]$$

X27

cevapoint of the orthocenter and the Clawson point

$$f(a,b,c)=\dfrac{(c^2+a^2-b^2)(a^2+b^2-c^2)}{b+c}$$

X28
クローソン点とX25のチェバ点
cevapoint of the Clawson point and X25

$$f(a,b,c)=\dfrac{a(c^2+a^2-b^2)(a^2+b^2-c^2)}{b+c}$$

X29

cevapoint of the incenter and the orthocenter

$$f(a,b,c)=\dfrac{(b+c-a)(c^2+a^2-b^2)(a^2+b^2-c^2)}{b+c}$$

X31

trilinear second power point

$$f(a,b,c)=a^3$$

X32

trilinear third power point

$$f(a,b,c)=a^4$$

X33

perspector of the orthic and intangents triangles

$$f(a,b,c)=a(b+c-a)(c^2+a^2-b^2)(a^2+b^2-c^2)$$

X34

orthocenter-beth conjugate of the orthocenter

$$f(a,b,c)=\dfrac{(c^2+a^2-b^2)(a^2+b^2-c^2)}{b+c-a}$$

X35
X36の内心外心調和共役点
incenter-circumcenter-harmonic conjugate of X36

$$f(a,b,c)=a^2(b^2+c^2-a^2+bc)$$

X36

inverse-in-circumcircle of the incenter

$$f(a,b,c)=a^2(b^2+c^2-a^2-bc)$$

X37

crosspoint of the incenter and the centroid

$$f(a,b,c)=a(b+c)$$

X38

crosspoint of the incenter and X75

$$f(a,b,c)=a(b^2+c^2)$$

X39
ブロカール中点
Brocard midpoint

$$f(a,b,c)=a^2(b^2+c^2)$$

X40
ベバン点
Bevan point

$$f(a,b,c)=a\left(\dfrac{b}{c+a-b}+\dfrac{c}{a+b-c}-\dfrac{a}{b+c-a}\right)$$

$$f_1(a,b,c)=a[a^3+a^2(b+c)-a(b+c)^2-(b+c)(b-c)^2]$$

X41

symmedian-point-Ceva conjugate of the trilinear second power point

$$f(a,b,c)=a^3(b+c-a)$$

X42

crosspoint of the incenter and the symmedian point

$$f(a,b,c)=a^2(b+c)$$

X43

symmedian-point-Ceva conjugate of the incenter

$$f(a,b,c)=a\left(\dfrac1b+\dfrac1c\right)-1$$

$$f_1(a,b,c)=a(ab+ac-bc)$$

X44

symmedian-point-line conjugate of the incenter

$$f(a,b,c)=a(b+c-2a)$$

X45

Mittenpunkt-beth conjugate of the incenter

$$f(a,b,c)=a(2b+2c-a)$$

X46

orthocenter-Ceva conjugate of the incenter

$$f(a,b,c)=a[b(c^2+a^2-b^2)+c(a^2+b^2-c^2)-a(b^2+c^2-a^2)]$$

X47
X34X110ベート共役点
X110-beth conjugate of X34

$$f(a,b,c)=a^3[(b^2+c^2-a^2)^2-2b^2c^2]$$

X48

crosspoint of the incenter and X63

$$f(a,b,c)=a^3(b^2+c^2-a^2)$$

X49

center of the sine-triple angle center

$$f(a,b,c)=a^4(b^2+c^2-a^2)[(b^2+c^2-a^2)^2-3b^2c^2]$$

X50
X184X74チェバ共役点
X74-Ceva conjugate of X184

$$f(a,b,c)=a^4[(b^2+c^2-a^2)^2-b^2c^2]$$

X51

centroid of the orthic triangle

$$f(a,b,c)=a^2[a^2(b^2+c^2)-(b^2-c^2)^2]$$

X52

orthocenter of the orthic triangle

$$f(a,b,c)=a^2[a^4-2a^2(b^2+c^2)+b^4+c^4][a^2(b^2+c^2)-(b^2-c^2)^2]$$

X53

symmedian point of the orthic triangle

$$f(a,b,c)=(c^2+a^2-b^2)(a^2+b^2-c^2)[a^2(b^2+c^2)-(b^2-c^2)^2]$$

X54
コスニタ点（九点円の中心の等角共役点）
Kosnita point (isogonal conjugate of the nine-point center)

$$f(a,b,c)=a^2[b^2(c^2+a^2)-(c^2-a^2)^2][c^2(a^2+b^2)-(a^2-b^2)^2]$$

X55

insimilicenter of the circumcenter and the incenter (isogonal conjugate of the Gergonne point)

$$f(a,b,c)=a^2(b+c-a)$$

X56

exsimilicenter of the circumcenter and the incenter (isogonal conjugate of the Nagel point)

$$f(a,b,c)=\dfrac{a^2}{b+c-a}$$

X57
ミッテンプンクトの等角共役点
isogonal conjugate of the mittenpunkt

$$f(a,b,c)=\dfrac{a}{b+c-a}$$

X58
シュピーカー心の等角共役点
isogonal conjugate of the Spieker center

$$f(a,b,c)=\dfrac{a^2}{b+c}$$

X59
フォイエルバッハ点の等角共役点
isogonal conjugate of the Feuerbach point

$$f(a,b,c)=\dfrac{a^2(c-a)^2(a-b)^2}{b+c-a}$$

X60

isogonal conjugate of the harmonic conjugate of the Feuerbach point with respect to the incenter and the nine-point center

$$f(a,b,c)=\dfrac{a^2(b+c-a)}{(b+c)^2}$$

X61

isogonal conjugate of the first Napoleon

$$f(a,b,c)=a^2(b^2+c^2-a^2+4\sqrt{3}\Delta)$$

X62

isogonal conjugate of the second Napoleon

$$f(a,b,c)=a^2(b^2+c^2-a^2-4\sqrt{3}\Delta)$$

X63
クローソン点の等角共役点
isogonal conjugate of the Clawson point

$$f(a,b,c)=a(b^2+c^2-a^2)$$

X64
ド・ロンシャン点の等角共役点
isogonal conjugate of the de Longchamps point

$$f(a,b,c)=a^2[a^4+2a^2(b^2-c^2)-(b^2-c^2)(3b^2+c^2)][a^4-2a^2(b^2-c^2)+(b^2-c^2)(b^2+3c^2)]$$

X65
ジェルゴンヌ三角形の垂心 （シフラー点の等角共役点）
orthocenter of the Gergonne triangle (isogonal conjugate of the Schiffler point)

$$f(a,b,c)=\dfrac{a(b+c)}{b+c-a}$$

X66
エクセター点の等角共役点
isogonal conjugate of the Exeter point

$$f(a,b,c)=(c^4+a^4-b^4)(a^4+b^4-c^4)$$

X67

isogonal conjugate of the far-out point

$$f(a,b,c)=(c^4+a^4-b^4-c^2a^2)(a^4+b^4-c^4-a^2b^2)$$

X68
プラソロフ点（元の三角形と垂足垂足三角形の透視点の等角共役点）
Prasolov point (isogonal conjugate of the perspector of the reference and orthic-of-orthic triangles)

$$f(a,b,c)=(b^2+c^2-a^2)[2c^2a^2-(c^2+a^2-b^2)^2][2a^2b^2-(a^2+b^2-c^2)^2]$$

X69

symmedian point of the anticomplementary triangle

$$f(a,b,c)=b^2+c^2-a^2$$

X70

isogonal conjugate of the circumcenter of the tangential triangle

$$f(a,b,c)=g(b,c,a)g(c,a,b)$$   ただし、 $$g(a,b,c)=b^4(c^2+a^2-b^2)^2+c^4(a^2+b^2-c^2)^2-a^4(b^2+c^2-a^2)^2+2a^2b^2c^2(b^2+c^2-a^2))$$

X71

isogonal conjugate of the cevapoint of the orthocenter and the Clawson point

$$f(a,b,c)=a^2(b+c)(b^2+c^2-a^2)$$

X72
X28の等角共役点（シュピーカー心のナーゲル点チェバ共役点）
isogonal conjugate of X28 (Nagel-point-Ceva conjugate of the Spieker center)

$$f(a,b,c)=a(b+c)(b^2+c^2-a^2)$$

X73

crosspoint of the incenter and the circumcenter (isotomic conjugate of the cevapoint of the incenter and the orthocenter)

$$f(a,b,c)=\dfrac{(b+c-a)(b^2+c^2-a^2)}{b+c}$$

X74
オイラー無限遠点の等角共役点（第一等力点と第二等力点のチェバ点）
isogonal conjugate of the Euler infinity point (cevapoint of the first and second isonynamic points)

$$f(a,b,c)=a^2g(b,c,a)g(c,a,b)$$  ただし、 $$g(a,b,c)=2a^4-a^2(b^2+c^2)-(b^2-c^2)^2$$

X75

isotomic conjugate of the incenter (isogonal conjugate of the trilinear second power point)

$$f(a,b,c)=\dfrac{1}{a}$$

X76

third Brocard point (isogonal conjugate of the trilinear third power point, isotomic conjugate of the symmedian point)

$$f(a,b,c)=\dfrac{1}{a^2}$$

X77

isogonal conjugate of the perspector of the orthic and intangents triangles

$$f(a,b,c)=\dfrac{a(b^2+c^2-a^2)}{b+c-a}$$

X78

isogonal conjugate of the orthocenter-beth conjugate of the orthocenter

$$f(a,b,c)=(b+c-a)(b^2+c^2-a^2)$$

X79
X35の等角共役点
isogonal conjugate of X35

$$f(a,b,c)=(c^2+a^2-b^2+ca)(a^2+b^2-c^2+ab)$$

X80

reflection of the incenter in the Feuerbach point (isogonal conjugate of X36, inverse-in-Fuhrmann-circle of the incenter)

$$f(a,b,c)=(c^2+a^2-b^2-ca)(a^2+b^2-c^2-ab)$$

X81

cevapoint of the incenter and the symmedian point (isogonal conjugate of X37

$$f(a,b,c)=\dfrac{a}{b+c}$$

X82
X38の等角共役点（内心と三線二乗点のチェバ点）
isogonal conjugate of X38 (cevapoint of the incenter and the trilinear second power point)

$$f(a,b,c)=\dfrac{a}{b^2+c^2}$$

X83

cevapoint of the centroid and the symmedian point (isogonal conjugate of X39

$$f(a,b,c)=\dfrac{1}{b^2+c^2}$$

X84
ベバン点の等角共役点
isogonal conjugate of the Bevan point

$$f(a,b,c)=ag(b,c,a)g(c,a,b)$$  ただし、  $$g(a,b,c)=\dfrac{b}{c+a-b}+\dfrac{c}{a+b-c}-\dfrac{a}{b+c-a}$$

$$f_1(a,b,c)=ag_1(b,c,a)g_1(c,a,b)$$  ただし、  $$g_1(a,b,c)=a^3+a^2(b+c)-a(b+c)^2-(b+c)(b-c)^2$$

X85
ミッテンプンクトの等距離共役点（X41の等角共役点）
isotomic conjugate of the mittenpunkt (isogonal conjugate of X41)

$$f(a,b,c)=\dfrac{1}{a(b+c-a)}$$

X86

cevapoint of the incenter and the centroid (isogonal conjugate of X42)

$$f(a,b,c)=\dfrac{1}{b+c-a}$$

X87

centroid-cross conjugate of the incenter (isogonal conjugate of X43)

$$f(a,b,c)=a(bc+ba-ca)(ca+cb-ab)$$

X88

isogonal conjugate of symmedian-point-line conjugate of the incenter

$$f(a,b,c)=a(c+a-2b)(a+b-2c)$$

X89

isogonal conjugate of the mittenpunkt-beth conjugate of the incenter

$$f(a,b,c)=a(2c+2a-b)(2a+2b-c)$$

X90

circumcenter-cross conjugate of the incenter (isogonal conjugate of X46)

$$f(a,b,c)=ag(b,c,a)g(c,a,b)$$  ただし、 $$g(a,b,c)=b(c^2+a^2-b^2)+c(a^2+b^2-c^2)-a(b^2+c^2-a^2)$$

X91
X47の等角共役点
isogonal conjugate of X47

$$f(a,b,c)=bc[(c^2+a^2-b^2)^2-2c^2a^2][(a^2+b^2-c^2)^2-2a^2b^2]$$

X92

cevapoint of the incenter and the Clawson point (isogonal conjugate of X48, isotomic conjugate of X63)

$$f(a,b,c)=bc(c^2+a^2-b^2)(a^2+b^2-c^2)$$

X93

isogonal conjugate of the center of the sine-triple angle center

$$f(a,b,c)=g(b,c,a)g(c,a,b)$$  ただし、 $$g(a,b,c)=a^2(b^2+c^2-a^2)[(b^2+c^2-a^2)^2-3b^2c^2]$$

X94
X50の等角共役点
isogonal conjugate of X50

$$f(a,b,c)=g(b,c,a)g(c,a,b)$$  ただし、 $$g(a,b,c)=a^2[(b^2+c^2-a^2)^2-b^2c^2]$$

X95

cevapoint of the centroid and the circumcenter (isogonal conjugate of X51, isotomic conjugate of the nine-point center)

$$f(a,b,c)=[b^2(c^2+a^2)-(c^2-a^2)^2][c^2(a^2+b^2)-(a^2-b^2)^2]$$

X96
X52の等角共役点（外心とプラソロフ点のチェバ点）
isogonal conjugate of X52 (cevapoint of the circumcenter and the Prasolov point)

$$f(a,b,c)=g(b,c,a)g(c,a,b)$$  ただし、 $$g(a,b,c)=[a^4-2a^2(b^2+c^2)+b^4+c^4][a^2(b^2+c^2)-(b^2-c^2)^2]$$

X97
X53の等角共役点
isogonal conjugate of X53

$$f(a,b,c)=a^2(b^2+c^2-a^2)g(b,c,a)g(c,a,b)$$  ただし、 $$g(a,b,c)=a^2(b^2+c^2)-(b^2-c^2)^2$$

X110
キーペルト放物線の焦点（シュタムラー双曲線の中心）
focus of the Kiepert parabola (center of the Stammler hyperbola)

$$f(a,b,c)=a^2(a^2-b^2)(a^2-c^2)$$

X115
キーペルト双曲線の中心（第一等角心と第二等角心の中点）
center of the Kiepert hyperbola (midpoint of the first and second isogonic points)

$$f(a,b,c)=(b^2-c^2)^2$$

X125
ジェラベック双曲線の中心
center of the Jerabek hyperbola

$$f(a,b,c)=(b^2+c^2-a^2)(b^2-c^2)^2$$

X140

midpoint of the circumcenter and the nine-point center (nine-point center of the medial triangle)

$$f(a,b,c)=2a^4-3a^2(b^2+c^2)+(b^2-c^2)^2$$

X142
ミッテンプンクトの補点（ジェルゴンヌ点とミッテンプンクトの中点）
complement of the mittenpunkt (midpoint of the Gergonne point and the mittenpunkt)

$$f(a,b,c)=a(b+c)-(b-c)^2$$

X179

first Ajima-Malfatti point

$$f(a,b,c)=\dfrac{1}{\left(2\sqrt{bc}+\sqrt{(a+b+c)(b+c-a)}\right)^2}$$

$$f(a,b,c)=u$$ $$\Longleftrightarrow$$ $$(c+a-b)^2(a+b-c)^2u^2-2(b^2+c^2-a^2+6bc)u+1=0$$ and $$0\lt u\lt \dfrac{4bc}{(c+a-b)^2(a+b-c)^2}$$

X182
ブロカール直径の中点（ブロカール円の中心、外心と類似重心の中点）
midpoint of the Brocard diameter (center of the Brocard circle, midpoint of the circumcenter and the symmedian point)

$$f(a,b,c)=a^2[a^2(b^2+c^2-a^2)+2b^2c^2]$$

X184
X125のブロカール円についての反転（外心と類似重心の交叉点）
inverse of X125 in the Brocard circle (crosspoint of the circumcenter and the symmedian point)

$$f(a,b,c)=a^4(b^2+c^2-a^2)$$

X187
シャウテ心（類似重心の外接円に関する反転）（第一等力点と第二等力点の中点）
Schoute center (inverse in the circumcircle of the symmedian point) (mid point of the first isodynamic point and the second isodynamic point)

$$f(a,b,c)=a^2(2a^2-b^2-c^2$$

X355
フールマン心
Fuhrmann center

$$f(a,b,c)=a^4-a^3(b+c)+2a^2bc+a(b+c)(b-c)^2-(b+c)^2(b-c)^2$$

X371

Kenmochi point (congruent squares point)

$$f(a,b,c)=a^2(b^2+c^2-a^2+4\Delta)$$   ただし、$$\Delta$$は元の三角形の面積

X372

circumcenter-symmedian-point-harmonic conjugate of the Kenmochi point

$$f(a,b,c)=a^2(b^2+c^2-a^2-4\Delta)$$   ただし、$$\Delta$$は元の三角形の面積

X381

midpoint of the centroid and the orthocenter

$$f(a,b,c)=a^4+a^2(b^2+c^2)-2(b^2-c^2)^2$$

X400
イフ・マルファッティ点
Yff-Malfatti point

$$f(a,b,c)=\dfrac{1}{\left(2\sqrt{bc}-\sqrt{(a+b+c)(b+c-a)}\right)^2}$$

$$f(a,b,c)=u$$ $$\Longleftrightarrow$$ $$(c+a-b)^2(a+b-c)^2u^2-2(b^2+c^2-a^2+6bc)u+1=0$$ and $$\dfrac{4bc}{(c+a-b)^2(a+b-c)^2}\lt u\lt \dfrac{16bc}{(c+a-b)^2(a+b-c)^2}$$

X483

radical center of the Ajima-Malfatti circles

$$f(a,b,c)=\dfrac{\sqrt{a}}{2\sqrt{bc}+\sqrt{(a+b+c)(b+c-a)}}$$

$$f(a,b,c)=u$$ $$\Longleftrightarrow$$ $$(c+a-b)^2(a+b-c)^2u^4-2a(b^2+c^2-a^2+6bc)u^2+a^2 = 0$$ and $$0 \lt u\lt \dfrac{2\sqrt{abc}}{(c+a-b)(a+b-c)}$$

$$f_1(a,b,c)=\dfrac{1}{2bc+\sqrt{bc(a+b+c)(b+c-a)}}$$

$$f_1(a,b,c)=u$$ $$\Longleftrightarrow$$ $$b^2c^2(c+a-b)^2(a+b-c)^2u^4-2bc(b^2+c^2-a^2+6bc)u^2+1 = 0$$ and $$0 \lt u\lt \dfrac{2}{(c+a-b)(a+b-c)}$$

X546

midpoint of the orthocenter and the nine-point center

$$f(a,b,c)=2a^4+a^2(b^2+c^2)-3(b^2-c^2)^2$$

X547

midpoint of the centroid and the nine-point center

$$f(a,b,c)=2a^4-7a^2(b^2+c^2)+5(b^2-c^2)^2$$

X548

midpoint of the nine-point center and the de Longchamps point

$$f(a,b,c)=-6a^4+5a^2(b^2+c^2)+(b^2-c^2)^2$$

X549

midpoint of the centroid and the circumcenter

$$f(a,b,c)=4a^4-5a^2(b^2+c^2)+(b^2-c^2)^2$$

X550

midpoint of the circumcenter and the de Longchamps point

$$f(a,b,c)=-4a^4+3a^2(b^2+c^2)+(b^2-c^2)^2$$

X551

midpoint of the incenter and the centroid

$$f(a,b,c)=4a+b+c$$

X560

trilinear fourth power point

$$f(a,b,c)=a^5$$

X561

isogonal conjugate of the trilinear fourth power point (isotomic conjugate of the trilinear second power point)

$$f(a,b,c)=\dfrac{1}{a^3}$$

X597

midpoint of the centroid and the symmedian point

$$f(a,b,c)=4a^2+b^2+c^2$$

X946

midpoint of the incenter and the orthocenter

$$f(a,b,c)=a^3(b+c)+(b-c)^2[a^2-a(b+c)-(b+c)^2]$$

X1001

midpoint of the incenter and mittenpunkt

$$f(a,b,c)=a[a^2-a(b+c)-2bc]$$

X1002
X1001の等角共役点
isogonal conjugate of X1001

$$f(a,b,c)=\dfrac{a}{a^2-a(b+c)-2bc}$$

X1142

first Malfatti-Rabinowitz point

$$f(a,b,c)=\dfrac{\left(2\sqrt{ca}+\sqrt{(a+b+c)(c+a-b)}\right)\left(2\sqrt{ab}+\sqrt{(a+b+c)(a+b-c)}\right)}{2\sqrt{bc}+\sqrt{(a+b+c)(b+c-a)}}-a$$

X1143

second Malfatti-Rabinowitz point

$$f(a,b,c)=\sqrt{b+c-a}\left(2\sqrt{bc}-\sqrt{(a+b+c)(b+c-a)}\right)$$

X1274

isotomic conjugate of the second Malfatti-Rabinowitz point (external second Malfatti-Rabinowitz point)

$$f(a,b,c)=\sqrt{b+c-a}\left(2\sqrt{bc}+\sqrt{(a+b+c)(b+c-a)}\right)$$

X1385

midpoint of the incenter and the circumcenter

$$f(a,b,c)=a[a(b^2+c^2-a^2)+(b+c-a)(c+a-b)(a+b-c)]$$

X1386

midpoint of the incenter and the symmedian point

$$f(a,b,c)=a[a(a+b+c)+(a^2+b^2+c^2)]$$

X1489

third Stevanović point

$$f(a,b,c)=\sqrt{a}\left(2\sqrt{bc}-\sqrt{(a+b+c)(b+c-a)}-\sqrt{(a-b+c)(a+b-c)}\right)$$

$$f_1(a,b,c)=a\left(2bc-\sqrt{bc(a+b+c)(b+c-a)}-\sqrt{bc(a-b+c)(a+b-c)}\right)$$

X1501

trilinear fifth power point

$$f(a,b,c)=a^6$$

X1917

trilinear sixth power point

$$f(a,b,c)=a^7$$

X1928

isogonal conjugate of the trilinear fourth power point (isotomic conjugate of the trilinear second power point)

$$f(a,b,c)=\dfrac{1}{a^5}$$

X2550
シュピーカー円と円(X4,2R)の内相似中心（ジェルゴンヌ点とナーゲル点の中点）
insimilicenter of the Spieker circle and the circle (X4,2R) (midpoint of the Gergonne point and the Nagel point)

$$f(a,b,c)=b+c+\dfrac{(c^2+a^2-b^2)(a^2+b^2-c^2)}{4abc}$$

$$f_1(a,b,c)=a^3-a^2(b+c)+a(b+c)^2-(b-c)^2(b+c)$$

X3082

radical center of the external Malfatti circles

$$f(a,b,c)=\dfrac{\sqrt{a}}{2\sqrt{bc}-\sqrt{(a+b+c)(b+c-a)}}$$

$$f(a,b,c)=u$$ $$\Longleftrightarrow$$ $$(c+a-b)^2(a+b-c)^2u^4-2a(b^2+c^2-a^2+6bc)u^2+a^2 = 0$$ and $$\dfrac{2\sqrt{abc}}{(c+a-b)(a+b-c)} \lt u\lt \dfrac{4\sqrt{abc}}{(c+a-b)(a+b-c)}$$

$$f_1(a,b,c)=\dfrac{1}{2bc-\sqrt{bc(a+b+c)(b+c-a)}}$$

$$f_1(a,b,c)=u$$ $$\Longleftrightarrow$$ $$b^2c^2(c+a-b)^2(a+b-c)^2u^4-2bc(b^2+c^2-a^2+6bc)u^2+1 = 0$$ and $$\dfrac{2}{(c+a-b)(a+b-c)} \lt u\lt \dfrac{4}{(c+a-b)(a+b-c)}$$

X3679
（重心とナーゲル点の中点）
(midpoint of the centroid and the Nagel point)

$$f(a,b,c)=2b+2c-a$$

X9233
P13U13の重心座標積（三線七乗点）
baricentric product of P13 and U13 (trilinear seventh power point)

$$f(a,b,c)=a^8$$

X12815

midpoint of the first and second Napoleon points

$$f(a,b,c)=4a^4-6a^2(b^2+c^2)+5(b^2-c^2)^2$$