[table][tr][td][url=https://www.geogebra.org/m/nzfg796n#material/zkmnwzkb][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]27.04.2023[/b][/i][/color])[/size][/td][/tr][/table][size=85][i][color=#ff00ff][right][/right][/color][/i][/size]

[size=85]A [b][color=#cc0000]2[/color][/b]-part [b][i][color=#ff7700]bicircular quartic[/color][/i][/b] possesses [b][color=#cc0000]4[/color][/b] different [b][i][color=#ff0000]concyclic[/color][/i][/b] [b][i][color=#00ff00]focal points [/color][/i][/b][math]f,f',f'',f'''[/math][br]and [b][color=#cc0000]4[/color][/b] paired [b][i][color=#0000ff]orthogonal[/color][/i][/b] [b][i][color=#bf9000]symmetry[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b]: [math]c_x[/math] (on which the [/size][size=85][b][i][color=#00ff00]focal points[/color][/i][/b][/size][size=85] lie) and [math]c_{y,}c_{E,}c_i[/math]; the latter is imaginary.[br][b][color=#cc0000]2[/color][/b] pairs of [b][i][color=#666666]double-touching[/color][/i][/b] [b][i][color=#ff0000]circles[/color][/i][/b] and a [b][i][color=#ff0000]hyperbolic[/color][/i][/b] [i][b][color=#ff0000]pencil[/color][/b][/i] of [b][i][color=#ff0000]circles[/color][/i][/b] around [b][color=#cc0000]2[/color][/b] of the [/size][size=85][b][i][color=#00ff00]focal points[/color][/i][/b][/size][size=85] create a [b][i][color=#ff7700]3-web-of-[/color][color=#ff0000]circles[/color][/i][/b],[br]if the [b][color=#cc0000]3[/color][/b] families of [b][i][color=#ff0000]circles[/color][/i][/b] belong to [b][color=#cc0000]3[/color][/b] different [b][i][color=#bf9000]symmetries[/color][/i][/b] with respect to the [b][i][color=#ff0000]circles[/color][/i][/b].[br][br]If one replaces the [b][i][color=#ff0000]hyperbolic pencil of circle[/color][/i][/b] by the [b][i][color=#0000ff]orthogonal [/color][color=#cc0000]elliptical pencil[/color][/i][/b], one obtains [br]also a [/size][size=85][b][i][color=#ff7700]3-web-of-[/color][color=#ff0000]circles[/color][/i][/b][/size][size=85].[/size]