Feuerbach Hyperbola by Steve Phelps

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2015-02-16 08:59

In collections A Tour of Triangle Geometry

This example demonstrates the Feuerbach hyperbola, which is the isogonal conjugate of the line through the incenter and the circumcenter. The center of the hyperbola is located at … Using GeoGebra commands such as Line, Circle, Center, Centroid, Mirror, LineBisector, Polygon, OrthogonalLine, Intersect, AngularBisector, Point, Locus, Midpoint, etc., the Feuerbach hyperbola can be constructed. Suitable for high school mathematics education and research.

GeoGebra Commands & Parameters

Line(A, B) → a
Line(B, C) → b
Line(C, A) → c
Circle(A, B, C) → d
Center(d) → O
Centroid(Polygon[A, B, C]) → G
Mirror(G, c) → G'
Mirror(G, a) → G'_1
Mirror(G, b) → G'_2
LineBisector(G'_1, G') → e
LineBisector(G', G'_2) → f
Polygon(A, B, C) → poly1, c_1, a_1, b_1
OrthogonalLine(A, b) → j
OrthogonalLine(B, c) → k
OrthogonalLine(C, a) → l
Intersect(l, j) → H
AngularBisector(B, A, C) → g
AngularBisector(C, B, A) → h
Intersect(g, h) → I
Line(I, O) → i
Point(i) → P
Mirror(P, c) → P'_3
Mirror(P, a) → P'_1
Mirror(P, b) → P'_2
LineBisector(P'_1, P'_3) → m
LineBisector(P'_3, P'_2) → n
Intersect(m, n) → P'
Locus(P', P) → loc1
Midpoint(B, C) → D
Midpoint(C, A) → E
Midpoint(A, B) → F_1
Circle(F_1, E, D) → p
OrthogonalLine(I, b) → q
Intersect(b, q) → J
Circle(I, J) → r
Intersect(p, r, 1) → F

AI Drawing Prompt Reference

Draw a triangle and find its incircle and circumcircle. Construct the perpendicular bisectors of the triangle's sides using the centroid as a reference point, and locate their intersections. Utilize these points to sketch the Feuerbach hyperbola, which represents the geometric relationship between the incircle and circumcircle. Finally, refine the shape of the hyperbola through mirroring and intersection operations.

The prompt above was compiled from the original drawing commands and is for reference only.