Section TarskiGe1.
Variable R : realFieldType.
Variable n : nat.
Implicit Types (a b c d : (@ Vector R n)).
Definition cong a b c d := (b - a) *m (b - a)^T == (d - c) *m (d - c)^T.
Definition bet a b c := betE a b c || betS a b c.
Definition extension a b k := k^-1 *: (b - a) + a.
Definition contraction a b k := k *: (b - a) + a.
Definition sqr_L2_norm_2D a b :=
(b 0 0 - a 0 0) ^+ 2 + (b 0 1 - a 0 1) ^+ 2.

Lemma upper_dim_dgc2 (a b c p q : 'rV[R]_(2)) (m := (1 / (1 + 1)) *: (p + q)) :
m 0 1 - p 0 1 == 0 -> p <> q ->
cong a p a q -> cong b p b q -> cong c p c q ->
(a 0 0 - b 0 0) * (b 0 1 - c 0 1) == (a 0 1 - b 0 1) * (b 0 0 - c 0 0).