Artzt parabolas of a triangle

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“…The results about areas are an immediate consequence. Also, since the circles , and (see Figure 13 in [1]) are circumcircles of these triangles, the results concerning these circles also follow at once.…”

“…The Artzt ellipse (see Figure 15 in [1]) has centre and passes through , and , as shown in Figure 3. Also shown is the Steiner ellipse , which is the unique ellipse with centre circumscribing the triangle .…”

“…On Artzt parabolas of a triangle [1]: Andrew Jobbings writes: The 'curve-stitching' construction (see Figures 1 and 3 in [1]) may be extended by continuing each sequence of points in both directions outside the original triangle , as shown in Figure 1. This defines the whole parabola, not just that part within triangle .…”

“…The points and parabolas are shown in Figure 2, where is the centroid of triangle . G A B C The two triangles shown in Figure 9 in [1] (which seems to be missing one of the parabolas) are the triangles and . P a P b P c M a M b M c Define the homothecies and to have centre and scale factors and 4 respectively.…”

“…On Constructing the 15th pentagon that tiles the plane [1]: Andrew Jobbings writes: The pentagonal tile may also be considered as two 'half squares' and two 'half equilateral triangles', of suitable sizes, placed together as shown in Figure 1. Figure 2 shows that this gives the correct lengths and angles.…”