Let TA = RB∩RC, and define TB and TC cyclically. Both of Euler's parents were very religious, his father being a pastor and his mother the daughter of a pastor.

This triangle is obtained by reflecting the circumcenter X3 about A, B, C. Reference: TCCT 6.13, p. 160. See Lucas triangles. G Denote B1 and C1 cyclically. (Reference: Preamble just before X8537), Following the construction of the 1st Ehrmann triangle, let A2 = BCBA∩CACB, and define B2 and C2 cyclically. The triangle bounded by the lines TabTac, TbcTba and TcaTcb is named the 3rd Jenkins triangle of ABC. Join LO and extend it so that it intersects the circumcircle at M. From I construct a perpendicular to AB, and let D be its foot, so ID = r. It is not difficult to prove that triangle ADI is similar to triangle MBL, so ID / BL = AI / ML, i.e.

AeBeCe and AiBiCi are the K798e & K798i triangles, respectively. ( Log Out /  https://www.khanacademy.org/.../triangle-property-review/v/euler-line The triangle A'B'C' whose Wasat triangle is ABC.

Let A2 = BCBA∩CACB, and define B2 and C2 cyclically.

La courbe est une ellipse si H est intérieur à ABC, et une hyperbole sinon (elle est même équilatère si H est sur le cercle circonscrit de ABC).

The centers of these squares are the vertices of the inner-squares triangles. Circles (A, P2), (B, P2) and (C, P2) intersect by pairs at P2, A", B", C".

A' = AX(56) ∩ perpendicular(X(7354), BC). Define MB, MC cyclically. Diderot had been invited to the court by Catherine the Great, but then annoyed her by trying to convert everyone to atheism. Reference: TCCT, p. 173. His name is associated with angles, approximation, circles, cycle, criterion, graphs, operator’s, polynomials, pseudo primes, … and Euler’s identity: said to be the most beautiful formula in the world.  Lucas antipodal: The tangents at Ap, Bp and Cp to the A-, B- and C-Lucas circles, respectively, bound the Lucas antipodal tangents triangle. (Reference: preamble just before X15015).

(Reference: X6044), A' is the A-center of curvature of the Steiner circumellipse, and similarly B' and C'. The triangle bounded by a", b" and c" is the Ursa-major triangle of ABC.

Note: If the tangents at B and C to the circumcircle of a triangle ABC intersect at A', then the circle with center A' passing through B and C is called A-excosine circle of ABC. The triangle A"B"C" is the 2nd Jenkins triangle of ABC.

r (Reference: X17817), Let A'B'C' be the orthic triangle of ABC.

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