From the general properties of homographies result the main properties of Steiner's ellipses of the triangle t':

1) From the invariance of cross-ratio and the properties of D, D' follows that the medians of t are mapped to the medians of t'.

2) This implies that the line at infinity is mapped to the line at infinity, hence the homography is an affinity.

3) Affinities, respect parallelity, length-ratios, area-ratios and map circles to ellipses.

4) This implies that the inner circle is mapped to the inner ellipse, that touches the sides of t' at their middles.

5) From (3) follows also that the tangent to the outer ellipse at the vertex is parallel to the opposite side of the triangle.

6) The centers of the two ellipses coincide with D' and the ellipses are homothetic with ratio 2 w.r. to that point.

7) The inner ellipse is the one inscribed with the maximum area, and the outer is the one circumscribed with the minimum area.

8) Since the affinity preserves area-ratios, it follows that

area(inner Steiner) = (pi/(3*sqrt(3)))*area(A'B'C') and

area(outer Steiner) = (4*pi/(3*sqrt(3)))area(A'B'C').

In fact, the ellipses of Steiner and the sides/vertices of the triangle intercept on the medians of the triangle equal segments. Thus, BH = HG = GE = EJ, CK = KG = GF = FI etc.

Thus, the conics can be constructed from five easy to find points through which pass (using the tool constructing a conic which passes through five points in general position).

IsotomicOfCircle.html

ProjectivityFixingVertices.html

SteinerPoint.html

TriangleCircumconics.html

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