HYACINTHOS 24036

 

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Ma, Mb, Mc = the midpoints of OA, OB, OC, resp.

Aa, Ab, Ac = the orthogonal projections of Ma on NA, NB, NC, resp.

Ba, Bb, Bc = the orthogonal projections of Mb on NA, NB, NC, resp.

Ca, Cb, Cc = the orthogonal projecitons of Mc on NA, NB, NC, resp.

M1, M2, M3 = the midpoints of NA, NB, NC, resp.

A1, A2, A3 = the orthogonal projections of M1 on OA, OB, OC, resp.

B1, B2, B3 = the orthogonal projections of M2 on OA, OB, OC, resp.

C1, C2, C3 = the orthogonal projecitons of M3 on OA, OB, OC, resp.

Oa, Ob, Oc = the circumcenters of the triangles AaAbAc, BaBbBc, CaCbCc, resp.

O1, O2, O3 = the circumcenters of the triangles A1A2A3, B1B2B3, C1C2C3, resp.

La, Lb, Lc = the Euler lines of OaO2O3, ObO3O1, OcO1O2, resp.

L1, L2, L3 = the Euler lines of O1ObOc, O2OcOa, O3OaOb, resp.

1. La, Lb, Lc are concurrent on the Euler line of ABC at the midpoint of ON.

2. L1, L2, L3 are concurrent on the Euler line of ABC at a point D such that OD / ON = 3/5

Is this point in ETC?

 

APΗ

[Angel Montesdeoca]:

 

1. La, Lb, Lc are concurrent on the Euler line of ABC at the midpoint of ON, X(140).

2. L1, L2, L3 are concurrent on the Euler line of ABC at a point D =5 O + 7 N = 7 G + O.

D = (10 a^4-17 a^2 (b^2+c^2)+7 (b^2-c^2)^2 : ... : ...),
with (6-9-13)-search number (3.14849305520217, 2.27066732812651, 0.615513383111177).

D is the   midpoint of X(i) and X(j) for these {i,j}:  {2,140},{3,5066},{546,8703},{ 547,549},{548,3845},{3534, 3853}.

D = reflection of X(3861) in X(5066).

D lies on these lines: {2,3}, {551,5844}, {952,3828}, {1698,3653}, {3054,5309}, {3055,7753}, {3582,7294}, {3584,5326}, {3624,3654}, {5418,6470}, {5420,6471}, {5650,5946}, {7749,9300}.

Angel M.