Four special Poncelet triangle families about the incircle

📝 Abstract

We describe four special families of ellipse-inscribed Poncelet triangles about the incircle which maintain certain triangle centers stationary and which also display interesting conservations.

💡 Analysis

We describe four special families of ellipse-inscribed Poncelet triangles about the incircle which maintain certain triangle centers stationary and which also display interesting conservations.

📄 Content

As special cases to [2], we introduce four special families of Poncelet triangles inscribed in an ellipse E and circumscribing their fixed incircle, let C be its center, i.e., the incenter X 1 (triangle center notation X k are after Kimberling [5]). Referring to Figure 1, the four Poncelet families are:

• focal-X 1 : the caustic is centered on a focus of E ; • iso-X 2 : caustic is centered on a special point on E ’s minor axis such that the barycenter X 2 is stationary;

• focal-X 4 : caustic is centered on a special point E ’s major axis such that the orthocenter X 4 is stationary on a focus of E ; • iso-X 7 : caustic is centered on a another special point on E ’s major axis such that the Gergonne point X 7 is stationary.

In the sections below, R, r, l i , i = 1, 2, 3 refer to a triangle’s circumradius, inradius, and sidelengths, respectively, see [14] for definitions.

Let E be given by (x/a) 2 + (y/b) 2 = 0. Referring to Figure 1 (top left): Proposition 1. The caustic centered at C 1 = [c, 0] admits a Poncelet family of triangles with r 1 given by:

Proof. This follows from [2, Prop. 2]: the radius r for a circular caustic (of Poncelet triangles) with center at [x c , y c ] is given by:

The focal-X 1 family is the polar image of the vertices of Chapple’s Porism (a porism of triangles interscribed between two circles [7]), with respect to the circumcircle, i.e., it is its tangential triangle. The former’s incenter X 1 coincides with the latter’s circumcenter X 3 .

Bicentric n-gons (a generalization of Chapple’s porism) conserve the sum of cosines of their internal angles θ i [10]. On the other hand, the polar family with respect to the circumcircle conserves ∑ n i=1 sin(θ i /2), for all n ≥ 3 [1, prop.26].

Proposition 2. For the focal-X 1 family, the sum of half-angle sines is invariant and given by:

Four families of ellipse-inscribed Poncelet triangles about the incircle. top-left: focal-X 1 (C at a focus of E ); top-right: iso-X 2 (C on minor axis of E , barycenter X 2 stationary); bot.-left: focal-X 4 (X 4 is a focus of E ); bot.-right: iso-X 7 (C on major axis of E , X 7 stationary). Video: youtu.be/nsHDfX6mA

Referring to Figure 1 (top right), it follows from the expressions for the center C 2 and semi-axis lengths a 2 , b 2 for the elliptic locus of the barycenter X 2 [2, Prop. 3] that: Proposition 3. For a circular caustic with center C 2 = 0, c b 2a and radius r 2 = b 2 , the barycenter X 2 is stationary at 0, c b 3a . Corollary 1. Over iso-X 2 triangles, Nagel’s point X 8 is stationary at E’s center.

Proof. Direct from the location of the stationary barycenter and the fact that

The Spieker center X 10 , incenter of the medial triangle and the midpoint of the incenter X 1 and Nagel’s point X 8 [5] will be stationary on E’s minor axis.

Observation 2. Since the barycenter X 2 is stationary, the family conserves |X 1 X 2 |, which for any triangle with sidelengths l i is given by [14,Triangle Centroid,eqn.11]:

Since foci of a triangle’s inconic are an isogonal conjugate pair [6] (see also Definition 2), the second focus must be the fixed orthocenter X 4 of the family (also within E ′ ), i.e., this is the MacBeath inconic.

Let K be the image of K ′ under A -1 . The the barycenter X 2 of the family in (E , K ) will be stationary because (i) by Lemma 1, the MacBeath’s centroid is stationary, and (ii) the centroid is the only triangle center which is equivariant with respect to affinities [5]. The barycenter X 2 must lie on OC because affinities preserve collinearities. Notice it will nevertheless not lie on the major axis of K (foci are not equivariant). □

Referring to Figure 3, experimental evidence suggests that for MacBeath-like Poncelet families of n-gons, n > 3, i.e., they are circle-inscribed and the caustic has a focus at the center of said circle, both vertex and area centroids C 0 and C 2 remain stationary on the caustic axis. When n = 4, C 0 lies at the caustic’s center. Corollary 3. Given an ellipse E centered at O and a point C in its interior, there is a caustic centered on C for which both c 0 and c 2 are stationary on OC. Indeed, over such families, the perimeter centroid, in general not expected to sweep a conic over Poncelet [11], is experimentally found to sweep an ellipse with major axis identical to the caustic’s.

Referring to Figure 1 (bottom left): Proposition 5. The orthocenter X 4 is stationary at a focus f = [±c, 0] of E for the circular caustic with center and radius given by:

Referring to Figure 4:

Observation 3. For the focal-X 4 family, the center (resp. other focus) of E is X 7952 (resp. X 18283 ). L 3 is an ellipse with a focus at E’s center. L 20 is twice as large, with a focus on E ’s distal focus.

Definition 3. A triangle’s polar circle is centered on the orthocenter X 4 and has squared radius given by [14, Polar circle]:

This quantity is positive (resp. negative) for obtuse (resp. acute triangles).

Proposition 6. The focal-X 4 family conserves the (neg

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