Prove that the points $Α,Β,Δ$ and $Ε$ are homocyclic

Question

$ΑΒΓ$ triangle with $ΑΔ⊥ΒΓ$ and $ΒΕ⊥ΑΓ.$

Prove that

(a) The points $Α,Β,Δ$ and $Ε$ are homocyclic.

(b) $ΔΕ∥(ε)$ where $(ε)$ is the tangent at $Γ$ of the surrounding circle of the triangle $ΑΒΓ$.

Answer 1

a) Since $ΑΔ⊥ΒΓ$ and $ΒΕ⊥ΑΓ$, the points Δ and E are on the same circle with AB as the diameter, hence $Α,Β,Δ$ and $Ε$ cyclic.

b) $\alpha = \angle ABΔ$ because they span the same arc. Since $\angle ABΔ + \angle AEΔ = 180$ due to cyclicality , the complimentary angle of $\alpha$ is equal to $\angle AEΔ $. Therefore EΔ || (ε).