Oriented angles and orthogonal projection.

This file proves lemmas relating to oriented angles involving orthogonal projections.

theorem EuclideanGeometry.oangle_self_orthogonalProjection {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p : P) {p' : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (hp : p ∉ s) (h : p' ∈ s) (hp' : p' ≠ ↑((orthogonalProjection s) p)) : oangle p (↑((orthogonalProjection s) p)) p' = ↑(Real.pi / 2) ∨ oangle p (↑((orthogonalProjection s) p)) p' = ↑(-Real.pi / 2)

theorem EuclideanGeometry.oangle_orthogonalProjection_self {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p : P) {p' : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (hp : p ∉ s) (h : p' ∈ s) (hp' : p' ≠ ↑((orthogonalProjection s) p)) : oangle p' (↑((orthogonalProjection s) p)) p = ↑(Real.pi / 2) ∨ oangle p' (↑((orthogonalProjection s) p)) p = ↑(-Real.pi / 2)

theorem EuclideanGeometry.two_zsmul_oangle_self_orthogonalProjection {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p : P) {p' : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (hp : p ∉ s) (h : p' ∈ s) (hp' : p' ≠ ↑((orthogonalProjection s) p)) : 2 • oangle p (↑((orthogonalProjection s) p)) p' = ↑Real.pi

theorem EuclideanGeometry.two_zsmul_oangle_orthogonalProjection_self {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p : P) {p' : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (hp : p ∉ s) (h : p' ∈ s) (hp' : p' ≠ ↑((orthogonalProjection s) p)) : 2 • oangle p' (↑((orthogonalProjection s) p)) p = ↑Real.pi