We introduce a concept "degree of triangle centers", and give a formula expressing the degree of triangle centers on generalized Euler lines. This generalizes the well known 2 : 1 point configuration on the Euler line. We also introduce a natural family of triangle centers based on the Ceva conjugate and the isotomic conjugate. This family contains many famous triangle centers, and we conjecture that the de- gree of triangle centers in this family always takes the form ( 2) k for some k 2 Z. MSC 2000: 51M05 (primary), 51A20 (secondary)

/pdf/degree-of-triangle-centers-and-a-generalization-of-the-euler-3blpn1jv01.pdf

Degree of Triangle Centers and a Generalization of the Euler Line

satisfying the tangent property implies the concurrence of the angle bisectors, say at the point O. So consider a hinged quadrilateral ABCD with points {X, Y, Z, W) satisfying the tangent condition. Then the angle bisectors meet at O. We can then drop perpendiculars from O to meet all of the sides internally (since each base angle in triangles of the form OAB are acute). Call these points {X\\ Y', Z', W'}; they clearly satisfy the tangent condition and are points of contact of the inscribed circle centred at O. What can we now say about {X, Y, Z, W}l Draw a circle, E, centred at O passing through X. Then a reflection on l\\ will carry Z onto itself and X into Y. Thus Y must also lie on £. Similarly Z and W are on Z. So each set of four points with the tangent property is concyclic, and all such quadruples lie on a circle centred at O, the meeting place of the angle bisectors. So we can complete the previous diagram by drawing the missing circles:

87.80 On a metric-related function for the triangle

86.08 An areal view of Feuerbach's theorem This theorem (1822, [1]) states that the nine-points circle of a triangle ABC is tangential to the inscribed circle and to each of the three escribed circles. In mis note we prove an equivalent theorem. Standard notation for the triangle is used as far as possible, and square brackets will be used for actual areal coordinates (sum normalised to unity) and round brackets for relative areal coordinates as in the Gazette note [2]. We first introduce the Nagel point of a triangle. This is the point of concurrence of the lines from die vertices to the corresponding points where the escribed circles touch the triangle (mid-perimeter points). In his Gazette note [3], Eperson uses vectors to show that the Nagel point (E, say) is coUinear with the centroid and the incentre. The actual areal coordinates of the Nagel point are easily found from Figure 2 in [3] to be ^[s-a,s-b,s-c] where 2s = a + b + c. The actual areal coordinates of the circumcentre O are R[a/bc, (51ca,ylab] where a, /?, y represent the quantities b + c a, c + a b , a + b c, respectively. These coordinates may be derived from the distances of O to the sides of triangle ABC, where for example the distance to BC is R cos A = Rallbc. An expression for the distance OE is now obtained by means of the metric

An areal view of Feuerbach's theorem

'-i \\ P which is equivalent to M[x, y, z] -> N[%(y + z), \\{z + x), \\{x + y)], clearly maps the vertices of the triangle onto the corresponding midpoints. It also maps the circumcircle onto the nine-points circle, the Nagel point E onto the incentre / (a, b, c) and the circle (E, 2r) to the incircle. Thus the internal part of the Feuerbach theorem is demonstrated. Furthermore, the image of F* is the Feuerbach point F, which may be shown under the linear mapping P to have relative areal coordinates

D. B. Eperson

Papers

80.12 The Euler line and the Eperson line