Definition contraction a b k := k *: (b - a) + a.

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