intersection0 a:C b:C -> X:P
  <- intersecting a b
  X <- intersection 0 a b
intersection0 a:L b:C -> X:P
  <- intersecting a b
  X <- intersection 0 a b
intersection0 a:C b:L -> X:P
  <- intersecting a b
  X <- intersection 0 b a
intersection1 a:C b:C -> X:P
  <- intersecting a b
  X <- intersection 1 a b
intersection1 a:L b:C -> X:P
  <- intersecting a b
  X <- intersection 1 a b
intersection1 a:C b:L -> X:P
  <- intersecting a b
  X <- intersection 1 b a
intersection_closer a:C b:C A:P -> X:P
  <- intersecting a b
  X <- intersection 0 a b A
intersection_closer a:L b:C A:P -> X:P
  <- intersecting a b
  X <- intersection 0 a b A
intersection_closer a:C b:L A:P -> X:P
  <- intersecting a b
  X <- intersection 0 b a A
intersection_remoter a:C b:C A:P -> X:P
  <- intersecting a b
  X <- intersection 1 a b A
intersection_remoter a:L b:C A:P -> X:P
  <- intersecting a b
  X <- intersection 1 a b A
intersection_remoter a:C b:L A:P -> X:P
  <- intersecting a b
  X <- intersection 1 b a A
intersections a:C b:C -> X0:P X1:P
  <- intersecting a b
  X0 <- intersection 0 a b
  X1 <- intersection 1 a b
intersections a:L b:C -> X0:P X1:P
  <- intersecting a b
  X0 <- intersection 0 a b
  X1 <- intersection 1 a b
intersections a:C b:L -> X0:P X1:P
  <- intersecting a b
  X0 <- intersection 0 b a
  X1 <- intersection 1 b a

compass A:P B:P C:P -> c:C
  d <- dist A B
  c <- circle C d
compass w:C X:P -> c:C
  r <- radius_of w
  c <- circle X r

circle C:P A:P -> w:C
  w <- compass A C C
  <- point_to_circle A w

tri_sides A:P B:P C:P -> a:L b:L c:L
  a <- line B C
  b <- line C A
  c <- line A B
tri_angles A:P B:P C:P -> alpha:A beta:A gamma:A
  alpha <- angle C A B
  beta <- angle A B C
  gamma <- angle B C A

concyclic A:P B:P C:P D:P ->
  w <- circumcircle A B C
  <- lies_on D w
collinear A:P B:P C:P ->
  l <- line A B
  <- lies_on C l

copy_triangle A0:P B0:P C0:P A1:P B1:P -> C1:P
  alpha beta gamma <- tri_angles A0 B0 C0
  dc1 <- direction_of A1 B1
  da1 <- angle_compute 0 dc1 1 beta 1
  db1 <- angle_compute 0 dc1 1 alpha -1
  a1 <- line_with_direction B1 da1
  b1 <- line_with_direction A1 db1
  C1 <- intersection a1 b1
  <- sim_aa A0 B0 C0 A1 B1 C1
copy_triangle_r A0:P B0:P C0:P A1:P B1:P -> C1:P
  alpha beta gamma <- tri_angles A0 B0 C0
  c1 <- line A1 B1
  dc1 <- direction_of A1 B1
  da1 <- angle_compute 0 dc1 1 beta -1
  db1 <- angle_compute 0 dc1 1 alpha 1
  a1 <- line_with_direction B1 da1
  b1 <- line_with_direction A1 db1
  C1 <- intersection a1 b1
  <- sim_aa_r A0 B0 C0 A1 B1 C1

sim_sss A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 B1 C1
  ra <- dist_ratio B0 C0 B1 C1
  rb <- dist_ratio C0 A0 C1 A1
  rc <- dist_ratio A0 B0 A1 B1
  <- == ra rb
  <- == ra rc
  THEN
  <- eq_angle C0 A0 B0 C1 A1 B1
  <- eq_angle B0 C0 A0 B1 C1 A1
  <- eq_angle A0 B0 C0 A1 B1 C1
  PROOF
  C1' <- copy_triangle A0 B0 C0 A1 B1
  wA <- circle A1 C1
  <- point_to_circle C1' wA
  wB <- circle B1 C1
  <- point_to_circle C1' wB
  <- == C1 C1'
sim_sas A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 B1 C1
  ra <- dist_ratio B0 C0 B1 C1
  rb <- dist_ratio C0 A0 C1 A1
  rc <- dist_ratio A0 B0 A1 B1
  <- == ra rc
  <- eq_angle A0 B0 C0 A1 B1 C1
  THEN
  <- == rb ra
  <- eq_angle C0 A0 B0 C1 A1 B1
  <- eq_angle B0 C0 A0 B1 C1 A1
  PROOF
  C1' <- copy_triangle A0 B0 C0 A1 B1
  lB <- line B1 C1
  <- lies_on C1' lB
  wB <- circle B1 C1
  <- point_to_circle C1' wB
  <- == C1 C1'
sim_Ssa A0:P B0:P C0:P A1:P B1:P C1:P ->
  <- oriented_as A0 B0 C0 A1 B1 C1
  c <- dist A0 B0
  a <- dist B0 C0
  <- dim_less a c
  ra <- dist_ratio B0 C0 B1 C1
  rb <- dist_ratio C0 A0 C1 A1
  rc <- dist_ratio A0 B0 A1 B1
  <- == rc ra
  <- eq_angle B0 C0 A0 B1 C1 A1
  THEN
  <- eq_angle A0 B0 C0 A1 B1 C1
  <- eq_angle C0 A0 B0 C1 A1 B1
  <- == ra rb
  PROOF
  A1' <- copy_triangle B0 C0 A0 B1 C1
  lC <- line C1 A1
  <- lies_on A1' lC
  wB <- circle B1 A1
  <- point_to_circle A1' wB
  <- == A1 A1'

sim_sss_r A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 C1 B1
  ra <- dist_ratio B0 C0 B1 C1
  rb <- dist_ratio C0 A0 C1 A1
  rc <- dist_ratio A0 B0 A1 B1
  <- == ra rb
  <- == ra rc
  THEN
  <- eq_angle C0 A0 B0 B1 A1 C1
  <- eq_angle A0 B0 C0 C1 B1 A1
  <- eq_angle B0 C0 A0 A1 C1 B1
  PROOF
  C1' <- copy_triangle_r A0 B0 C0 A1 B1
  wA <- circle A1 C1
  <- point_to_circle C1' wA
  wB <- circle B1 C1
  <- point_to_circle C1' wB
  <- == C1 C1'
sim_sas_r A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 C1 B1
  ra <- dist_ratio B0 C0 B1 C1
  rb <- dist_ratio C0 A0 C1 A1
  rc <- dist_ratio A0 B0 A1 B1
  <- == ra rc
  <- eq_angle A0 B0 C0 C1 B1 A1
  THEN
  <- == rb ra
  <- eq_angle C0 A0 B0 B1 A1 C1
  <- eq_angle B0 C0 A0 A1 C1 B1
  PROOF
  C1' <- copy_triangle_r A0 B0 C0 A1 B1
  lB <- line B1 C1
  <- lies_on C1' lB
  wB <- circle B1 C1
  <- point_to_circle C1' wB
  <- == C1 C1'
sim_Ssa_r A0:P B0:P C0:P A1:P B1:P C1:P ->
  <- oriented_as A0 B0 C0 A1 C1 B1
  c <- dist A0 B0
  a <- dist B0 C0
  <- dim_less a c
  ra <- dist_ratio B0 C0 B1 C1
  rb <- dist_ratio C0 A0 C1 A1
  rc <- dist_ratio A0 B0 A1 B1
  <- == rc ra
  <- eq_angle B0 C0 A0 A1 C1 B1
  THEN
  <- eq_angle A0 B0 C0 C1 B1 A1
  <- eq_angle C0 A0 B0 B1 A1 C1
  <- == ra rb
  PROOF
  A1' <- copy_triangle_r B0 C0 A0 B1 C1
  lC <- line C1 A1
  <- lies_on A1' lC
  wB <- circle B1 A1
  <- point_to_circle A1' wB
  <- == A1 A1'

cong_sss A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 B1 C1
  <- eq_dist A0 B0 A1 B1
  <- eq_dist B0 C0 B1 C1
  <- eq_dist C0 A0 C1 A1
  THEN
  <- eq_angle C0 A0 B0 C1 A1 B1
  <- eq_angle A0 B0 C0 A1 B1 C1
  <- eq_angle B0 C0 A0 B1 C1 A1
  PROOF
  <- sim_sss A0 B0 C0 A1 B1 C1
cong_sas A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- not_collinear A0 B0 C0
  <- eq_dist A0 B0 A1 B1
  <- eq_angle A0 B0 C0 A1 B1 C1
  <- eq_dist B0 C0 B1 C1
  THEN
  <- eq_angle C0 A0 B0 C1 A1 B1
  <- eq_dist C0 A0 C1 A1
  <- eq_angle B0 C0 A0 B1 C1 A1
  PROOF
  <- sim_sas A0 B0 C0 A1 B1 C1
cong_asa A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 B1 C1
  <- eq_angle C0 A0 B0 C1 A1 B1
  <- eq_dist A0 B0 A1 B1
  <- eq_angle A0 B0 C0 A1 B1 C1
  THEN
  <- eq_dist B0 C0 B1 C1
  <- eq_angle B0 C0 A0 B1 C1 A1
  <- eq_dist C0 A0 C1 A1
  PROOF
  <- sim_aa A0 B0 C0 A1 B1 C1
cong_Ssa A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 B1 C1
  c <- dist A0 B0
  a <- dist B0 C0
  <- dim_less a c
  <- eq_dist A0 B0 A1 B1
  <- eq_dist B0 C0 B1 C1
  <- eq_angle B0 C0 A0 B1 C1 A1
  THEN
  <- eq_angle A0 B0 C0 A1 B1 C1
  <- eq_angle C0 A0 B0 C1 A1 B1
  <- eq_dist C0 A0 C1 A1
  PROOF
  <- sim_Ssa A0 B0 C0 A1 B1 C1

cong_sss_r A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 C1 B1
  <- eq_dist A0 B0 A1 B1
  <- eq_dist B0 C0 B1 C1
  <- eq_dist C0 A0 C1 A1
  THEN
  <- eq_angle C0 A0 B0 B1 A1 C1
  <- eq_angle A0 B0 C0 C1 B1 A1
  <- eq_angle B0 C0 A0 A1 C1 B1
  PROOF
  <- sim_sss_r A0 B0 C0 A1 B1 C1
cong_sas_r A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 C1 B1
  <- eq_dist A0 B0 A1 B1
  <- eq_angle A0 B0 C0 C1 B1 A1
  <- eq_dist B0 C0 B1 C1
  THEN
  <- eq_angle C0 A0 B0 B1 A1 C1
  <- eq_dist C0 A0 C1 A1
  <- eq_angle B0 C0 A0 A1 C1 B1
  PROOF
  <- sim_sas_r A0 B0 C0 A1 B1 C1
cong_asa_r A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- not_collinear A0 B0 C0
  <- eq_angle C0 A0 B0 B1 A1 C1
  <- eq_dist A0 B0 A1 B1
  <- eq_angle A0 B0 C0 C1 B1 A1
  THEN
  <- eq_dist B0 C0 B1 C1
  <- eq_angle A0 B0 C0 C1 B1 A1
  <- eq_dist C0 A0 C1 A1
  PROOF
  <- sim_aa_r A0 B0 C0 A1 B1 C1
cong_Ssa_r A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- oriented_as A0 B0 C0 A1 C1 B1
  c <- dist A0 B0
  a <- dist B0 C0
  <- dim_less a c
  <- eq_dist A0 B0 A1 B1
  <- eq_dist B0 C0 B1 C1
  <- eq_angle B0 C0 A0 A1 C1 B1
  THEN
  <- eq_angle A0 B0 C0 C1 B1 A1
  <- eq_angle C0 A0 B0 B1 A1 C1
  <- eq_dist C0 A0 C1 A1
  PROOF
  <- sim_Ssa_r A0 B0 C0 A1 B1 C1

isosceles_aa A:P B:P C:P -> 
  <- not_collinear A B C
  <- eq_angle A B C B C A
  THEN
  <- eq_dist A B A C
  PROOF
  <- sim_aa_r C A B B A C
equilateral_a A:P B:P C:P ->
  <- not_collinear A B C
  <- eq_angle A B C B C A
  <- eq_angle A B C C A B
  THEN
  <- eq_dist A B A C
  <- eq_dist A B B C
  PROOF
  <- sim_aa A B C B C A
equilateral_s A:P B:P C:P ->
  <- eq_dist A B A C
  <- eq_dist A B B C
  THEN
  <- eq_angle A B C B C A
  <- eq_angle A B C C A B
  PROOF
  <- sim_sss A B C B C A

midpoint_uq M:P A:P B:P -> 
  l <- line A B
  <- lies_on M l
  <- not_eq A B
  <- eq_dist A M M B
  M' <- midpoint A B
  THEN
  <- == M M'
  PROOF
  <- point_to_perp_bisector M A B

parallel l0:L l1:L -> 
  ang <- angle l0 l1
  <- angle_pred 0 ang 1
parallel l0:L A1:P B1:P -> 
  ang <- angle l0 A1 B1
  <- angle_pred 0 ang 1
parallel A0:P B0:P l1:L -> 
  ang <- angle A0 B0 l1
  <- angle_pred 0 ang 1
parallel A0:P B0:P A1:P B1:P -> 
  ang <- angle A0 B0 A1 B1
  <- angle_pred 0 ang 1

paraline l:L A:P -> pa:L
  dir <- direction_of l
  pa <- line_with_direction A dir
  THEN
  <- parallel l pa
  PROOF
paraline X0:P X1:P A:P -> pa:L
  l <- line X0 X1
  pa <- paraline l A

perpendicular l0:L l1:L ->
  ang <- angle l0 l1
  <- angle_pred -1/2 ang 1
perpendicular l0:L A1:P B1:P ->
  ang <- angle l0 A1 B1
  <- angle_pred -1/2 ang 1
perpendicular A0:P B0:P l1:L ->
  ang <- angle A0 B0 l1
  <- angle_pred -1/2 ang 1  
perpendicular A0:P B0:P A1:P B1:P -> 
  ang <- angle A0 B0 A1 B1
  <- angle_pred -1/2 ang 1  

foot A:P l:L -> F:P
  p <- perpline l A
  F <- intersection p l
foot A:P X0:P X1:P -> F:P
  l <- line X0 X1
  F <- foot A l
foot A:P w:C -> F:P
  r <- radius_of w
  O <- center_of w
  dir <- half_direction O A
  F <- double_direction O dir r
  THEN
  <- lies_on F w
  PROOF
  <- point_to_circle F w

dist A:P l:L -> d:D
  <- not_on A l
  F <- foot A l
  d <- dist A F
dist l:L A:P -> d:D
  d <- dist A l
eq_dist A0:P l0:L A1:P l1:L ->
  d0 <- dist A0 l0
  d1 <- dist A1 l1
  <- == d0 d1

double_direction_inv A:P B:P -> dir:A d:D
  d <- dist A B
  dir <- half_direction A B
  B' <- double_direction A dir d
  THEN
  <- == B B'
  PROOF
  w <- circle A B
  <- point_to_circle B' w
  
copy_vector A:P B:P C:P -> D:P
  dir d <- double_direction_inv A B
  D <- double_direction C dir d

reflect A:P M:P -> B:P
  B <- copy_vector A M M
  M' <- midpoint A B
  _ _ <- double_direction_inv M A
  THEN
  <- == M M'
  PROOF
  <- midpoint_uq M A B
reflect A:P l:L -> B:P
  M <- foot A l
  B <- reflect A M
  THEN
  M' <- midpoint A B
  l' <- perp_bisector A B
  <- == M M'
  <- == l l'
  PROOF

opposite_point A:P w:C -> B:P
  <- lies_on A w
  O <- center_of w
  B <- reflect A O
  THEN
  <- lies_on B w
  r <- radius_of w
  d <- dist A O
  <- == r d
  PROOF
  <- point_on_circle A w
  <- point_to_circle B w

isosceles_trapezoid A':P A:P B:P -> B':P
  p <- perp_bisector A A'
  B' <- reflect B p
  THEN
  <- eq_dist A B A' B'
  dir_ab <- direction_of A B
  dir_ab' <- direction_of A' B'
  dir_p <- direction_of p
  <- angle_pred 0 dir_ab 1 dir_ab' 1 dir_p -2
  PROOF
  M <- midpoint A A'
  <- point_on_perp_bisector M B B'
  <- cong_sas_r A M B A' M B'
reflected_concurrent A:P A':P B:P B':P -> X:P
  p <- perp_bisector A A'
  pb <- perp_bisector B B'
  <- == p pb
  ab <- line A B
  ab' <- line A' B'
  X <- intersection p ab
  <- point_on_perp_bisector X A A'
  <- point_on_perp_bisector X B B'
  <- lies_on X ab'

midsegment A:P B:P C:P -> C':P B':P
  C' <- midpoint A B
  B' <- midpoint C A
  THEN
  <- parallel B C B' C'
  r <- dist_ratio B' C' B C
  <- ratio_pred 2 r 1
  PROOF
  <- sim_sas B A C C' A B'

parallelogram_aa A:P B:P C:P D:P ->
  <- not_collinear A B C
  <- parallel A B C D
  <- parallel B C D A
  THEN
  <- eq_dist A B C D
  <- eq_dist B C D A
  PROOF
  <- cong_asa A C B C A D
parallelogram_ss A:P B:P C:P D:P -> 
  <- oriented_as A B C C D A
  <- eq_dist A B C D
  <- eq_dist B C D A
  THEN
  <- parallel A B C D
  <- parallel B C D A
  PROOF
  <- cong_sss A B C C D A
parallelogram_sa A:P B:P C:P D:P -> 
  <- oriented_as A B C C D A
  <- eq_dist A B C D
  <- parallel A B C D
  THEN
  <- eq_dist B C D A
  <- parallel B C D A
  PROOF
  <- cong_sas C A B A C D

rectangle_axis A:P B:P C:P D:P -> l:L
  <- perpendicular A B B C
  <- perpendicular B C C D
  <- perpendicular C D D A
  M0 <- midpoint A B
  M1 <- midpoint C D
  l <- line M0 M1
  THEN
  l0 <- perp_bisector A B
  l1 <- perp_bisector C D
  <- == l0 l
  <- == l1 l
  PROOF
  <- parallelogram_aa A B C D
  <- parallelogram_sa M0 B C M1
  <- perpendicular M0 M1 A B

circumcenter A:P B:P C:P -> O:P
  <- not_collinear A B C
  a <- perp_bisector B C
  b <- perp_bisector C A
  c <- perp_bisector A B
  w <- circumcircle A B C
  O <- center_of w
  THEN
  <- lies_on O a
  <- lies_on O b
  <- lies_on O c
  <- eq_dist O A O B
  <- eq_dist O A O C
  r <- radius_of w
  <- eq_dist O A r
  PROOF
  <- point_on_circle A w
  <- point_on_circle B w
  <- point_on_circle C w
  <- point_to_perp_bisector O A B
  <- point_to_perp_bisector O B C
  <- point_to_perp_bisector O C A

cong_circumradius A0:P B0:P C0:P A1:P B1:P C1:P -> 
  <- cong_sss A0 B0 C0 A1 B1 C1
  w0 <- circumcircle A0 B0 C0
  w1 <- circumcircle A1 B1 C1
  r0 <- radius_of w0
  r1 <- radius_of w1
  THEN
  <- == r0 r1
  PROOF
  O0 <- circumcenter A0 B0 C0
  O1 <- copy_triangle A0 B0 O0 A1 B1
  <- cong_sas O0 A0 C0 O1 A1 C1
  w1' <- circle O1 A1
  <- point_to_circle B1 w1'
  <- point_to_circle C1 w1'

circle_pos_of A:P w:C -> pos:A
  <- lies_on A w
  O <- center_of w
  pos <- half_direction O A
of_circle_pos pos:A w:C -> A:P
  O <- center_of w
  r <- radius_of w
  A <- double_direction O pos r
arc_length A:P B:P w:C -> al:A
  pA <- circle_pos_of A w
  pB <- circle_pos_of B w
  al <- angle_compute 0 pA -1 pB 1
eq_arcs A:P B:P C:P D:P w:C ->
  AB <- arc_length A B w
  CD <- arc_length C D w
  <- == AB CD

eq_arcs_to_eq_dist A:P B:P C:P D:P w:C ->
  <- eq_arcs A B C D w
  THEN
  <- eq_dist A B C D
  PROOF
  <- point_on_circle A w
  <- point_on_circle B w
  <- point_on_circle C w
  <- point_on_circle D w
  O <- center_of w
  <- cong_sas A O B C O D
eq_dist_to_eq_arcs A:P B:P C:P D:P w:C ->
  <- lies_on A w
  <- lies_on B w
  <- lies_on C w
  <- lies_on D w
  <- eq_dist A B C D
  O <- center_of w
  <- oriented_as O A B O C D
  THEN
  <- eq_arcs A B C D w
  PROOF
  <- point_on_circle A w
  <- point_on_circle B w
  <- point_on_circle C w
  <- point_on_circle D w
  <- cong_sss O A B O C D

secant_direction A:P B:P w:C -> d:A
  <- lies_on A w
  <- lies_on B w
  pa <- circle_pos_of A w
  pb <- circle_pos_of B w
  d <- direction_of A B
  THEN
  <- angle_pred 1/2 pa 1 pb 1 d -1
  PROOF
  O <- center_of w
  <- point_on_circle A w
  <- point_on_circle B w
  <- isosceles_ss O A B

inscribed_angle X:P Y:P Z:P -> ins_angle:A
  w <- circumcircle X Y Z
  C <- center_of w
  ins_angle <- angle X Y Z
  al <- arc_length X Z w
  THEN
  <- == ins_angle al
  PROOF
  _ <- secant_direction Y X w
  _ <- secant_direction Y Z w
on_circle_by_angle X:P Y:P A:P w:C ->
  <- lies_on X w
  <- lies_on Y w
  ang <- arc_length X Y w
  ang' <- angle X A Y
  <- == ang ang'
  THEN
  <- lies_on A w
  PROOF
  y <- line Y A
  A' <- intersection_remoter y w Y
  _ <- inscribed_angle X A' Y

concyclic_to_angles X:P Y:P A:P B:P ->
  <- concyclic X Y A B
  THEN
  <- eq_angle X A Y X B Y
  PROOF
  alpha <- inscribed_angle X A Y
  beta <- inscribed_angle X B Y
  <- == alpha beta
angles_to_concyclic X:P Y:P A:P B:P -> 
  <- eq_angle X A Y X B Y
  THEN
  <- concyclic X Y A B
  PROOF
  w <- circumcircle X A Y
  _ <- inscribed_angle X A Y
  <- on_circle_by_angle X Y B w

diacircle A:P B:P -> w:C
  M <- midpoint A B
  w <- circle M A
  THEN
  <- lies_on A w
  <- lies_on B w
  PROOF
  <- point_to_circle A w
  <- point_to_circle B w

point_on_diacircle X:P A:P B:P ->
  <- not_eq A X
  <- not_eq B X
  c <- diacircle A B
  <- lies_on X c
  THEN
  <- perpendicular A X X B
  PROOF
  _ <- inscribed_angle A X B
point_to_diacircle X:P A:P B:P ->
  <- perpendicular A X X B
  M <- midpoint A B
  w <- diacircle A B
  THEN
  <- lies_on X w
  <- eq_dist M A M X
  PROOF
  _ _ <- midsegment A B X
  <- point_on_perp_bisector M A X
  <- point_to_circle X w

is_tangent w:C l:L -> 
  O <- center_of w
  X <- foot O l
  <- lies_on X w
is_tangent l:L w:C ->
  <- is_tangent w l

touchpoint w:C l:L -> X:P
  <- is_tangent w l
  THEN
  O <- center_of w
  X <- foot O l
  <- lies_on X l
  PROOF
touchpoint l:L w:C -> X:P
  X <- touchpoint w l

tangent_at A:P w:C -> t:L
  <- lies_on A w
  O <- center_of w
  l <- line A O
  t <- perpline l A
  THEN
  <- is_tangent w t
  PROOF

tangent0 A:P w:C -> l:L X:P
  O <- center_of w
  d <- diacircle A O
  X <- intersection0 w d
  l <- line A X
  THEN
  <- is_tangent l w
  PROOF
  <- point_on_diacircle X A O
tangent1 A:P w:C -> l:L X:P
  O <- center_of w
  d <- diacircle A O
  X <- intersection1 w d
  l <- line A X
  THEN
  <- is_tangent l w
  PROOF
  <- point_on_diacircle X A O
tangents A:P w:C -> t1:L X1:P t2:L X2:P
  t1 X1 <- tangent0 A w
  t2 X2 <- tangent1 A w
tangent_closer A:P w:C P:P -> l:L X:P
  O <- center_of w
  d <- diacircle A O
  X <- intersection_closer w d P
  l <- line A X
  THEN
  <- is_tangent l w
  PROOF
  <- point_on_diacircle X A O
tangent_remoter A:P w:C P:P -> l:L X:P
  O <- center_of w
  d <- diacircle A O
  X <- intersection_remoter w d P
  l <- line A X
  THEN
  <- is_tangent l w
  PROOF
  <- point_on_diacircle X A O

deltoid A:P B:P C:P D:P -> 
  <- isosceles_ss A B D
  <- eq_dist B C C D
  THEN
  <- eq_angle D A C C A B
  <- eq_angle B C A A C D
  <- perpendicular A C B D
  PROOF
  <- cong_sss_r D A C B A C
chord_is_perp w0:C w1:C ->
  X0 X1 <- intersections w0 w1
  O0 <- center_of w0
  O1 <- center_of w1
  THEN
  <- perpendicular O0 O1 X0 X1
  PROOF
  <- point_on_circle X0 w0
  <- point_on_circle X0 w1
  <- point_on_circle X1 w0
  <- point_on_circle X1 w1
  <- deltoid O0 X0 O1 X1

angle_bisector_int A:P B:P C:P -> l:L
  l1 <- line B A
  l2 <- line B C
  d1 <- half_direction B A
  d2 <- half_direction B C
  dir <- angle_compute 0 d1 1 d2 1
  l <- line_with_direction B dir
angle_bisector_ext A:P B:P C:P -> l:L
  l1 <- line B A
  l2 <- line B C
  d1 <- half_direction A B
  d2 <- half_direction B C
  dir <- angle_compute 0 d1 1 d2 1
  l <- line_with_direction B dir

point_on_angle_bisector X:P l1:L l2:L ->
  A <- intersection l1 l2
  l <- line A X
  <- eq_angle l1 l l l2
  THEN
  <- eq_dist X l1 X l2
  PROOF
  F1 <- foot X l1
  F2 <- foot X l2
  <- cong_asa_r A X F1 A X F2
point_to_angle_bisector X:P l1:L l2:L ->
  A <- intersection l1 l2
  l <- line A X
  <- eq_dist X l1 X l2
  THEN
  <- eq_angle l1 l l l2
  PROOF
  F1 <- foot X l1
  F2 <- foot X l2
  <- cong_Ssa_r A X F1 A X F2

midpoint_op_arc A:P B:P C:P -> M:P
  l <- angle_bisector_int A B C
  w <- circumcircle A B C
  M <- intersection_remoter l w B
  THEN
  <- eq_arcs A M M C w
  <- eq_dist A M M C
  PROOF
  _ <- secant_direction B A w
  _ <- secant_direction B M w
  _ <- secant_direction B C w
  <- eq_arcs_to_eq_dist A M M C w
midpoint_arc A:P B:P C:P -> M:P
  M_op <- midpoint_op_arc A B C
  w <- circumcircle A B C
  M <- opposite_point M_op w
  l <- angle_bisector_ext A B C
  THEN
  <- eq_arcs A M M C w
  <- eq_dist A M M C
  <- lies_on M l
  PROOF
  _ <- secant_direction B A w
  _ <- secant_direction B M w
  _ <- secant_direction B C w
  <- eq_arcs_to_eq_dist B M M C w
midpoint_arc A:P B:P w:C -> M:P
  <- lies_on A w
  <- lies_on B w
  dir <- half_direction A B
  r <- radius_of w
  O <- center_of w
  dir2 <- angle_compute -1/4 dir 1
  M <- double_direction O dir2 r
  THEN
  <- lies_on M w
  <- eq_arcs A M M B w
  PROOF
  <- point_on_circle A w
  <- point_on_circle B w
  <- point_to_circle M w
  _ <- secant_direction A B w

incenter A:P B:P C:P -> I:P
  <- not_collinear A B C
  ia <- angle_bisector_int C A B
  ib <- angle_bisector_int A B C
  ic <- angle_bisector_int B C A
  I <- intersection ia ib
  a b c <- tri_sides A B C
  THEN
  <- lies_on I ia
  <- lies_on I ib
  <- lies_on I ic
  <- eq_dist I a I b
  <- eq_dist I b I c
  PROOF
  <- point_on_angle_bisector I b c
  <- point_on_angle_bisector I c a
  <- point_to_angle_bisector I a b
incircle A:P B:P C:P -> i:C
  I <- incenter A B C
  a b c <- tri_sides A B C
  F <- foot I a
  i <- circle I F
  THEN
  <- is_tangent a i
  <- is_tangent b i
  <- is_tangent c i
  PROOF
  Fb <- foot I b
  Fc <- foot I c
  <- point_to_circle Fb i
  <- point_to_circle Fc i

excenter A:P B:P C:P -> E:P
  <- not_collinear A B C
  ia <- angle_bisector_int C A B
  eb <- angle_bisector_ext A B C
  ec <- angle_bisector_ext B C A
  E <- intersection ia eb
  a b c <- tri_sides A B C
  THEN
  <- lies_on E ia
  <- lies_on E eb
  <- lies_on E ec
  <- eq_dist E a E b
  <- eq_dist E b E c
  PROOF
  <- point_on_angle_bisector E b c
  <- point_on_angle_bisector E c a
  <- point_to_angle_bisector E a b
excircle A:P B:P C:P -> e:C
  E <- excenter A B C
  a b c <- tri_sides A B C
  F <- foot E a
  e <- circle E F
  THEN
  <- is_tangent a e
  <- is_tangent b e
  <- is_tangent c e
  PROOF
  Fb <- foot E b
  Fc <- foot E c
  <- point_to_circle Fb e
  <- point_to_circle Fc e

median A:P B:P C:P -> m:L
  M <- midpoint B C
  m <- line A M
centroid A:P B:P C:P -> G:P
  a <- median A B C
  b <- median B C A
  c <- median C A B
  Ma <- midpoint B C
  Mb <- midpoint C A
  Mc <- midpoint A B
  G <- intersection b c
  THEN
  <- lies_on G a
  <- lies_on G b
  <- lies_on G c
  PROOF
  _ _ <- midsegment A B C
  <- sim_aa B C G Mb Mc G
  _ _ <- midsegment B C A
  <- sim_sas G C A G Mc Ma

altitude A:P B:P C:P -> p:L
  p <- perpline B C A
orthocenter A:P B:P C:P -> H:P
  a <- altitude A B C
  b <- altitude B C A
  c <- altitude C A B
  H <- intersection a b
  THEN
  <- lies_on H a
  <- lies_on H b
  <- lies_on H c
  PROOF
  aa <- paraline B C A
  bb <- paraline C A B
  cc <- paraline A B C
  AA <- intersection bb cc
  BB <- intersection cc aa
  CC <- intersection aa bb
  <- cong_asa A C B C A BB
  <- cong_asa B A C A B CC
  <- cong_asa C B A B C AA
  <- midpoint_uq A BB CC
  <- midpoint_uq B CC AA
  <- midpoint_uq C AA BB
  H' <- circumcenter AA BB CC
  <- == H H'

reflected_ortocenter A:P B:P C:P -> H:P H':P w:C
  H <- orthocenter A B C
  w <- circumcircle A B C
  a <- line B C
  H' <- reflect H a
  THEN
  <- lies_on H' w
  PROOF
  hb <- altitude B A C
  ha <- altitude C A B
  <- point_on_perp_bisector B H H'
  <- point_on_perp_bisector C H H'
  <- angles_to_concyclic B C A H'

circle9 A:P B:P C:P -> f:C
  Ma <- midpoint B C
  Mb <- midpoint C A
  Mc <- midpoint A B
  f <- circumcircle Ma Mb Mc
  Fa <- foot A B C
  Fb <- foot B C A
  Fc <- foot C A B
  H <- orthocenter A B C
  Ha <- midpoint H A
  Hb <- midpoint H B
  Hc <- midpoint H C
  THEN
  <- lies_on Ma f
  <- lies_on Mb f
  <- lies_on Mc f
  <- lies_on Fa f
  <- lies_on Fb f
  <- lies_on Fc f
  <- lies_on Ha f
  <- lies_on Hb f
  <- lies_on Hc f
  PROOF
  _ _ <- midsegment A B C
  _ _ <- midsegment H B C
  _ _ <- midsegment B H A
  _ _ <- midsegment C H A
  _ <- rectangle_axis Mb Mc Hb Hc
  _ _ <- midsegment B C A
  _ _ <- midsegment H C A
  _ _ <- midsegment C H B
  _ _ <- midsegment A H B
  _ <- rectangle_axis Mc Ma Hc Ha
  fH <- circumcircle Ha Hb Hc
  _ _ <- midsegment C A B
  _ _ <- midsegment H A B
  <- cong_circumradius Ma Mb Mc Ha Hb Hc
  F <- circumcenter Ma Mb Mc
  FH <- circumcenter Ha Hb Hc
  <- == FH F
  <- == fH f

  <- point_to_diacircle Fa A B
  <- point_to_diacircle Fa C A
  <- cong_sss_r Mb Mc Fa Mc Mb Ma
  <- angles_to_concyclic Mb Mc Ma Fa
  <- point_to_diacircle Fb B C
  <- point_to_diacircle Fb A B
  <- cong_sss_r Mc Ma Fb Ma Mc Mb
  <- angles_to_concyclic Mc Ma Mb Fb
  <- point_to_diacircle Fc C A
  <- point_to_diacircle Fc B C
  <- cong_sss_r Ma Mb Fc Mb Ma Mc
  <- angles_to_concyclic Ma Mb Mc Fc

isogonal_by_refl X:P A:P B:P C:P -> l:L
  _ b c <- tri_sides A B C
  Xb <- reflect X b
  Xc <- reflect X c
  l <- perp_bisector Xb Xc
  x <- line X A
  THEN
  <- lies_on A l
  <- eq_angle b x l c
  PROOF
  <- point_on_perp_bisector A X Xb
  <- point_on_perp_bisector A X Xc
  <- isosceles_ss A Xb Xc
  <- point_to_perp_bisector A Xb Xc
conjugate X:P A:P B:P C:P -> Y:P
  a b c <- tri_sides A B C
  Xa <- reflect X a
  Xb <- reflect X b
  Xc <- reflect X c
  Y <- circumcenter Xa Xb Xc
  THEN
  <- eq_angle X A B C A Y
  <- eq_angle X B C A B Y
  <- eq_angle X C A B C Y
  PROOF
  ia <- isogonal_by_refl X A B C
  ib <- isogonal_by_refl X B C A
  ic <- isogonal_by_refl X C A B

inverse A:P w:C -> A':P
  r <- radius_of w
  O <- center_of w
  dir d <- double_direction_inv O A
  d' <- ratio_compute 1 r 2 d -1
  A' <- double_direction O dir d'
  A'' <- double_direction O dir d

quadrilateral_PQ A:P B:P C:P D:P -> P:P Q:P
  ab <- line A B
  cd <- line C D
  P <- intersection ab cd
  bc <- line B C
  da <- line D A
  Q <- intersection bc da

miquel A:P B:P C:P D:P -> M:P
  P Q <- quadrilateral_PQ A B C D
  wBC <- circumcircle C B P
  wDA <- circumcircle D A P
  wAB <- circumcircle A B Q
  wCD <- circumcircle C D Q
  M <- intersection_remoter wDA wBC P
  THEN
  <- lies_on M wAB
  <- lies_on M wCD
  PROOF
  <- concyclic_to_angles M C P B
  <- concyclic_to_angles M D P A
  <- angles_to_concyclic M Q B A
  <- concyclic_to_angles M B P C
  <- concyclic_to_angles M A P D
  <- angles_to_concyclic Q M D C

concyclic_to_eq_power X:P A:P B:P C:P D:P ->
  <- collinear X A B
  <- collinear X C D
  <- not_eq A B
  <- not_eq C D
  <- concyclic A B C D
  da <- dist X A
  db <- dist X B
  dc <- dist X C
  dd <- dist X D
  pow_ab <- ratio_compute 1 da 1 db 1
  pow_cd <- ratio_compute 1 dc 1 dd 1
  THEN
  <- == pow_ab pow_cd
  PROOF
  <- concyclic_to_angles A C B D
  <- concyclic_to_angles B D A C
  <- sim_aa_r X A D X C B

polar A:P w:C -> a:L
  A' <- inverse A w
  O <- center_of w
  a <- perpline O A' A'
pole a:L w:C -> A:P
  O <- center_of w
  A' <- foot O a
  A <- inverse A' w

point_on_perp_bisector X:P p:L A:P B:P ->
  p' <- perp_bisector A B
  <- == p p'
  <- point_on_perp_bisector X A B
point_to_perp_bisector X:P p:L A:P B:P ->
  p' <- perp_bisector A B
  <- == p p'
  <- point_to_perp_bisector X A B