angle_bisector x a b c
x : a b c x
a b c = ncoll a b c
x : eqangle b a b x b x b c
bisect a b c

angle_mirror x a b c
x : a b c x
a b c = ncoll a b c
x : eqangle b a b c b c b x
amirror a b c

circle x a b c
x : a b c
a b c = ncoll a b c
x : cong x a x b, cong x b x c
bline a b, bline a c

circumcenter x a b c
x : a b c
a b c = ncoll a b c
x : cong x a x b, cong x b x c
bline a b, bline a c

eq_quadrangle a b c d
d : a b c d
 =
a : ; b : ; c : ; d : cong d a b c
eq_quadrangle

eq_trapezoid a b c d
d : a b c
 =
a : ; b : ; c : ; d : para d c a b, cong d a b c
eq_trapezoid

eq_triangle x b c
x : b c
b c = diff b c
x : cong x b b c, cong b c c x; eqangle b x b c c b c x, eqangle x c x b b x b c
circle b b c, circle c b c

eqangle2 x a b c
x : a b c x
a b c = ncoll a b c
x : eqangle a b a x c x c b
eqangle2 a b c

eqdia_quadrangle a b c d
d : a b c d
 =
a : ; b : ; c : ; d : cong d b a c
eqdia_quadrangle

eqdistance x a b c
x : a b c x
a b c = diff b c
x : cong x a b c
circle a b c

foot x a b c
x : a b c
a b c = ncoll a b c
x : perp x a b c, coll x b c
tline a b c, line b c

free a
a : a
 =
a :
free

incenter x a b c
x : a b c
a b c = ncoll a b c
x : eqangle a b a x a x a c, eqangle c a c x c x c b; eqangle b c b x b x b a
bisect a b c, bisect b c a

incenter2 x y z i a b c
i : a b c, x : i b c, y : i c a, z : i a b
a b c = ncoll a b c
i : eqangle a b a i a i a c, eqangle c a c i c i c b; eqangle b c b i b i b a; x : coll x b c, perp i x b c; y : coll y c a, perp i y c a; z : coll z a b, perp i z a b; cong i x i y, cong i y i z
incenter2 a b c

excenter x a b c
x : a b c
a b c = ncoll a b c
x : eqangle a b a x a x a c, eqangle c a c x c x c b; eqangle b c b x b x b a
bisect b a c, exbisect b c a

excenter2 x y z i a b c
i : a b c, x : i b c, y : i c a, z : i a b
a b c = ncoll a b c
i : eqangle a b a i a i a c, eqangle c a c i c i c b; eqangle b c b i b i b a; x : coll x b c, perp i x b c; y : coll y c a, perp i y c a; z : coll z a b, perp i z a b; cong i x i y, cong i y i z
excenter2 a b c

centroid x y z i a b c
x : b c, y : c a, z : a b, i : a x b y
a b c = ncoll a b c
x : coll x b c, cong x b x c; y : coll y c a, cong y c y a; z : coll z a b, cong z a z b; i : coll a x i, coll b y i; coll c z i
centroid a b c

ninepoints x y z i a b c
x : b c, y : c a, z : a b, i : x y z
a b c = ncoll a b c
x : coll x b c, cong x b x c; y : coll y c a, cong y c y a; z : coll z a b, cong z a z b; i : cong i x i y, cong i y i z
ninepoints a b c

intersection_cc x o w a
x : o w a
o w a = ncoll o w a
x : cong o a o x, cong w a w x
circle o o a, circle w w a

intersection_lc x a o b
x : a o b
a o b = diff a b, diff o b, nperp b o b a
x : coll x a b, cong o b o x
line b a, circle o o b

intersection_ll x a b c d
x : a b c d
a b c d = npara a b c d, ncoll a b c d
x : coll x a b, coll x c d
line a b, line c d

intersection_lp x a b c m n
x : a b c m n
a b c m n = npara m n a b, ncoll a b c, ncoll c m n
x : coll x a b, para c x m n
line a b, pline c m n

intersection_lt x a b c d e
x : a b c d e
a b c d e = ncoll a b c, nperp a b d e
x : coll x a b, perp x c d e
line a b, tline c d e

intersection_pp x a b c d e f
x : a b c d e f
a b c d e f = diff a d, npara b c e f
x : para x a b c, para x d e f
pline a b c, pline d e f

intersection_tt x a b c d e f
x : a b c d e f
a b c d e f = diff a d, npara b c e f
x : perp x a b c, perp x d e f
tline a b c, tline d e f

iso_triangle a b c
c : a b c
 =
a : ; b : ; c : eqangle b a b c c b c a, cong a b a c
isos

lc_tangent x a o
x : x a o
a o = diff a o
x : perp a x a o
tline a a o

midpoint x a b
x : a b
a b = diff a b
x : coll x a b, cong x a x b
midp a b

mirror x a b
x : a b
a b = diff a b
x : coll x a b, cong b a b x
pmirror a b

nsquare x a b
x : a b
a b = diff a b
x : cong x a a b, perp x a a b
rotaten90 a b

on_aline x a b c d e
x : x a b c d e
a b c d e = ncoll c d e
x : eqangle a x a b d c d e
aline e d c b a

on_aline2 x a b c d e
x : x a b c d e
a b c d e = ncoll c d e
x : eqangle x a x b d c d e
aline2 e d c b a

on_bline x a b
x : x a b
a b = diff a b
x : cong x a x b, eqangle a x a b b a b x
bline a b

on_circle x o a
x : x o a
o a = diff o a
x : cong o x o a
circle o o a

on_line x a b
x : x a b
a b = diff a b
x : coll x a b
line a b

on_pline x a b c
x : x a b c
a b c = diff b c, ncoll a b c
x : para x a b c
pline a b c

on_tline x a b c
x : x a b c
a b c = diff b c
x : perp x a b c
tline a b c

orthocenter x a b c
x : a b c
a b c = ncoll a b c
x : perp x a b c, perp x b c a; perp x c a b
tline a b c, tline b c a

parallelogram a b c x
x : a b c
a b c = ncoll a b c
x : para a b c x, para a x b c; cong a b c x, cong a x b c
pline a b c, pline c a b

pentagon a b c d e

 =
a : ; b : ; c : ; d : ; e :
pentagon

psquare x a b
x : a b
a b = diff a b
x : cong x a a b, perp x a a b
rotatep90 a b

quadrangle a b c d

 =
a : ; b : ; c : ; d :
quadrangle

r_trapezoid a b c d
d : a b c
 =
a : ; b : ; c : ; d : para a b c d, perp a b a d
r_trapezoid

r_triangle a b c
c : a b c
 =
a : ; b : ; c : perp a b a c
r_triangle

rectangle a b c d
c : a b c , d : a b c
 =
a : ; b : ; c : perp a b b c ; d : para a b c d, para a d b c; perp a b a d, cong a b c d, cong a d b c, cong a c b d
rectangle

reflect x a b c
x : a b c
a b c = diff b c, ncoll a b c
x : cong b a b x, cong c a c x; perp b c a x
reflect a b c

risos a b c
c : a b
 =
a : ; b : ; c : perp a b a c, cong a b a c; eqangle b a b c c b c a
risos

s_angle a b x y
x : a b x
a b = diff a b
x : s_angle a b x y
s_angle a b y

segment a b

 =
a : ; b :
segment

shift x b c d
x : b c d
b c d = diff d b
x : cong x b c d, cong x c b d
shift d c b

square a b x y
x : a b, y : a b x
a b = diff a b
x : perp a b b x, cong a b b x; y : para a b x y, para a y b x; perp a y y x, cong b x x y, cong x y y a, perp a x b y, cong a x b y
square a b

isquare a b c d
c : a b , d : a b c
 =
a : ; b : ; c : perp a b b c, cong a b b c; d : para a b c d, para a d b c; perp a d d c, cong b c c d, cong c d d a, perp a c b d, cong a c b d
isquare

trapezoid a b c d
d : a b c d
 =
a : ; b : ; c : ; d : para a b c d
trapezoid

triangle a b c

 =
a : ; b : ; c :
triangle

triangle12 a b c
c : a b c
 =
a : ; b : ; c : rconst a b a c 1 2
triangle12

2l1c x y z i a b c o
x : a b c o y z i, y : a b c o x z i, z : a b c o x y i, i : a b c o x y z
a b c o = cong o a o b, ncoll a b c
x y z i : coll x a c, coll y b c, cong o a o z, coll i o z, cong i x i y, cong i y i z, perp i x a c, perp i y b c
2l1c a b c o

e5128 x y a b c d
x : a b c d y, y : a b c d x
a b c d = cong c b c d, perp b c b a
x y : cong c b c x, coll y a b, coll x y d, eqangle a b a d x a x y
e5128 a b c d

3peq x y z a b c
z : b c z , x : a b c z y, y : a b c z x
a b c = ncoll a b c
z : coll z b c ; x y : coll x a b, coll y a c, coll x y z, cong z x z y
3peq a b c

trisect x y a b c
x : a b c y, y : a b c x
a b c = ncoll a b c
x y : coll x a c, coll y a c, eqangle b a b x b x b y, eqangle b x b y b y b c
trisect a b c

trisegment x y a b
x : a b y, y : a b x
a b = diff a b
x y : coll x a b, coll y a b, cong x a x y, cong y x y b
trisegment a b

on_dia x a b
x : x a b
a b = diff a b
x : perp x a x b
dia a b

ieq_triangle a b c
c : a b
 =
a : ; b : ; c : cong a b b c, cong b c c a; eqangle a b a c c a c b, eqangle c a c b b c b a
ieq_triangle

on_opline x a b
x : x a b
a b = diff a b
x : coll x a b
on_opline a b

cc_tangent0 x y o a w b
x : o a w b y, y : o a w b x
o a w b = diff o a, diff w b, diff o w
x y : cong o x o a, cong w y w b, perp x o x y, perp y w y x
cc_tangent0 o a w b

cc_tangent x y z i o a w b
x : o a w b y, y : o a w b x, z : o a w b i, i : o a w b z
o a w b = diff o a, diff w b, diff o w
x y : cong o x o a, cong w y w b, perp x o x y, perp y w y x; z i : cong o z o a, cong w i w b, perp z o z i, perp i w i z
cc_tangent o a w b

eqangle3 x a b d e f
x : x a b d e f
a b d e f = ncoll d e f, diff a b, diff d e, diff e f
x : eqangle x a x b d e d f
eqangle3 a b d e f

tangent x y a o b
x y : o a b
a o b = diff o a, diff o b, diff a b
x : cong o x o b, perp a x o x; y : cong o y o b, perp a y o y
tangent a o b

on_circum x a b c
x : a b c
a b c = ncoll a b c
x : cyclic a b c x
cyclic a b c