Circumcenter

Playing with the circumcenter

Given the triangle t = ABC, its circumcenter is the intersection point O of the medial lines OJ, OD, OF of its sides BC, CA and AB respectively. There are some interesting relations between the medial lines and the symmedians, leading to a couple of construction-methods of a symmedian of t. This is explained in the remarks below.

Given the triangle t = ABC, consider the intersection points G, H of the medial lines FO, DO with sides AC, AB respectively.
1) q = BCGH is a cyclic quadrilateral (angle(HCG) = angle(HBG) = angle(A)).
2) The circumcircle c of q passes through the circumcenter O of the triangle t.
3) The circumcircle d of AFD and c intersect at O and E, such that A, E, I are on a line. I being the intersection of the medial of BC and c, diametral to O (orthogonality of OE to line AE).
4) angle(A) = angle(BGI) = angle(BHI) = angle(BEI) = angle(IEC) = angle(IGC) and p = AHIG is a parallelogram.
5) HG is bisected by AI at M, hence AI is the symmedian w.r. to A (see Antiparallels.html ).
6) angle(FAE) = angle(FOE) = angle(GCE) and since, because of the symmedian AI, angle(JAC) = angle(FAE) the triangle ANC is isosceles and N is on the medial of AC.
7) Similarly APB is isosceles and P is on the medial line of AB. Hence E is also the intersection point of the sides BP and CN of the isosceli ABP, ACN, defined through the median AJ.
8) angle(IBC) = angle(ICB) = angle(A) leads to an easy construction of the symmedian AI.
9) angle(BAE) = angle(ECA), angle(ABE) = angle(EAC) shows that ABE and CAE are similar triangles.

For an application of these remarks in a case of determination of the focus of a parabola, look at the file ParabolaSkew.html .

Playing with the circumcenter

See Also