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Steiner's Hat ∆ (purple, top cusp not shown) is tangent to E at the intersections Qi of the Apollonius Hyperbola H (olive green) with E, excluding M . Notice H passes through the center of E. Left: When M is exterior to the evolute E * (dashed purple), only one tangent Q1 is present. Right: When M is interior to E * , three tangent points Qi, i = 1, 2, 3 arise. The intersections of E * are given in Equation 10. Note: the area ratio of ∆-to-E * is always 4/3.

Steiner's Hat ∆ (purple, top cusp not shown) is tangent to E at the intersections Qi of the Apollonius Hyperbola H (olive green) with E, excluding M . Notice H passes through the center of E. Left: When M is exterior to the evolute E * (dashed purple), only one tangent Q1 is present. Right: When M is interior to E * , three tangent points Qi, i = 1, 2, 3 arise. The intersections of E * are given in Equation 10. Note: the area ratio of ∆-to-E * is always 4/3.

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The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.