Second proof. Let O be the circumcenter of ABC and let ω be the circle centered at D with radius DB. Let lines AB and AC meet ω at P and Q, respectively. Since ∠PBQ = ∠DQC + ∠BAC = 1 2(∠BDC + ∠DOC) = 90 ◦, we see that PQ is a diameter of ω and hence passes through D. Since ∠ABC = ∠AQP and ∠ACB = ∠APQ, we see that triangles ABC and AQP are similar. If M is the midpoint of BC, noting that D is th...

Euclidean geometry, as presented by Euclid, consists of straightedge-andcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathematics” leads to the development of a first-order theory ECG of “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rathe...