[table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/vgq8qxrw][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAA2CAYAAABA3FA2AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAACpSURBVGhD7dkxCsJAFEXR/wZiJWJhIW7MUnApriwLEFdhZy0iiN8M2tjdLr94h8wEUt3yQaRhyMiMiH7mpulRv0vU/Gm/dymOohxFOYpyFOUoylGUoygdd9tye0qv1upFTUVenoSjKEdRjqIcRTmKchTlKErX/abgyLvUG3nKs5cn4ijKUZSjKEdRjqIcRRWN6j9six2dxkMu9YiV7t+Ps8l45iJu73V8AE/fHKUjFbbZAAAAAElFTkSuQmCC[/img][/url][/td][td][size=50] this activity is a page of [color=#980000][i][b]geogebra-book[/b][/i][/color][br] [url=https://www.geogebra.org/m/y9cj4aqt][color=#0000ff][u][i][b]elliptic functions & bicircular quartics & . . .[/b][/i][/u][/color][/url]([color=#ff7700][i][b]12.02.2023[/b][/i][/color])[/size][/td][/tr][/table]

[size=85]Midpoint [b][i][color=#ff7700]conic sections[/color][/i][/b] have [b][color=#cc0000]3[/color][/b] families of [b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b]:[br]one of these families is in the interior, which is the [b][i][color=#ff7700]conic section[/color][/i][/b] side containing the [b][i][color=#00ff00]foci[/color][/i][/b].[br]The two pairs on the outside are the [math]y[/math]-axisymmetrical [/size][size=85][b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b][/size][br][size=85]and the [b][i][color=#444444]tangents[/color][/i][/b]. We count the [/size][size=85][b][i][color=#444444]tangents[/color][/i][/b][/size][size=85] among the [/size][size=85][b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85], since they [br]are [b][i][color=#0000ff]möbius-geometrically[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b] through [math]\infty[/math] - and [math]\infty[/math] can be interpreted as both a [b][i][color=#ff7700]curve point[/color][/i][/b] and a [b][i][color=#00ff00]focal point[/color][/i][/b].[br]We count the set of [/size][size=85][b][i][color=#444444]tangents[/color][/i][/b][/size][size=85] twice: for a [b]2[/b]-part [b][i][color=#ff7700]bicircular quartic[/color][/i][/b] let [b][color=#cc0000]2[/color][/b] of the [b][i][color=#00ff00]foci[/color][/i][/b] coindent.[br]If this limit point is chosen as [math]\infty[/math] , then [b][color=#cc0000]2[/color][/b] families of [/size][size=85][b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85] merge into the group of [b][i][color=#444444]tangents[/color][/i][/b], [br]the corresponding [b][i][color=#0000ff]directices circles[/color][/i][/b] merge into the [b][i][color=#0000ff]directrix-circle[/color][/i][/b] of the [/size][size=85][b][i][color=#444444]tangents[/color][/i][/b][/size][size=85].[br]This clarifies a little the fact that the [/size][size=85][b][i][color=#444444]tangents[/color][/i][/b][/size][size=85] (double-counted) and the [/size][size=85][b][i][color=#999999]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b][/size][size=85] on the outside [br]can create a [b][i][color=#ff7700]3-web-of-circles[/color][/i][/b]. [/size]

[size=85][b][color=#cc0000]2[/color][/b] [b][i][color=#444444]double-touching[/color][/i] [i][color=#ff0000]circles[/color][/i][/b] pass through each point on the outside of a midpoint [b][i][color=#ff7700]conic section[/color][/i][/b].[br]However, no [b][i][color=#ff7700]3-web-of-circles[/color][/i][/b] can be constructed from [b]2[/b] of these [b][i][color=#ff0000]circles[/color][/i][/b] and one [b][i][color=#444444]tangent[/color][/i][/b] per [b][i][color=#ff0000]point[/color][/i][/b].[br]In general, the following applies to [b][i][color=#ff7700]2-part quartics[/color][/i][/b]: the [b][color=#cc0000]3[/color][/b] [b][i][color=#999999]double-touching [/color][color=#ff0000]circles[/color][/i][/b] per point can only be [br]be extended to a [/size][size=85][b][i][color=#ff7700]3-web-of-circles[/color][/i][/b][/size][size=85], if they belong to different [b][i][color=#bf9000]symmetries[/color][/i][/b]![/size]