Orthopole and Dilation of Circumcircle

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2017-08-20 08:35

This example demonstrates the dynamic geometric relationship between the orthopole and the dilation of the circumcircle in a triangle. It involves core mathematical concepts such as dilation, secant line or secant, median line, logarithmic functions, limits, and parametric curves. The construction utilizes GeoGebra commands and objects like Line, OrthogonalLine, Intersect, Dilate, Midpoint, Circle, Point, Tangent, and Locus. It is suitable for high school and above students to explore the connection between geometric transformations and algebraic relationships.

GeoGebra Commands & Parameters

Line(C, A) → h
OrthogonalLine(B, h) → r
Line(B, C) → g
OrthogonalLine(A, g) → q
Intersect(q, r) → I
Dilate(C, a, I) → C'
Dilate(A, a, I) → A'
Dilate(B, a, I) → B'
Midpoint(B, C) → M
Midpoint(C, A) → N
Midpoint(A, B) → L
Line(A, B) → f
Circle(A', B', C') → c
Point(c) → D
Tangent(D, c) → i,
OrthogonalLine(A, i) → j
OrthogonalLine(B, i) → k
OrthogonalLine(C, i) → l
Intersect(i, j) → E
Intersect(i, l) → F
Intersect(i, k) → G
OrthogonalLine(E, g) → m
OrthogonalLine(F, f) → n
OrthogonalLine(G, h) → p
Intersect(m, n) → H
Locus(H, D) → loc1

AI Drawing Prompt Reference

Draw a triangle ABC and construct its circumcircle. Find the midpoints M, N, L of A, B, C respectively, and connect A and B to form line f. With I as the center, perform a similarity transformation on A, B, and C along a line a to get A', B', and C', and construct a circle c with A', B', and C' as its center. Choose a point D on circle c, draw a tangent i from D to circle c, and construct the perpendiculars j, k, and l to i, intersecting i at points E, F, and G. Then, construct the perpendiculars m, n, and p to E, F, and G with respect to BC, CA, and AB, respectively, and their intersection H is the desired point. Finally, the locus of H with respect to D is loc1.

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