Reflection of a Point on the Circumcircle

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2015-02-15 02:37

Triangle Circle Transformation

In collections A Tour of Triangle Geometry

This example demonstrates that the reflections of a point on the circumcircle of a triangle over the sides of the triangle are collinear. Key mathematical concepts include circles, reflection transformations, and collinearity. The construction uses GeoGebra commands such as Line, Circle, Point, OrthogonalLine, Intersect, and Mirror. Suitable for middle school and above students or teachers.

GeoGebra Commands & Parameters

Line(A, B) → a
Line(B, C) → b
Line(C, A) → c
Circle(A, B, C) → d
Point(d) → P
OrthogonalLine(P, a) → e
OrthogonalLine(P, c) → f
OrthogonalLine(P, b) → g
Intersect(a, e) → P_c
Intersect(c, f) → P_b
Intersect(b, g) → P_a
Line(P_c, P_b) → h
Mirror(P, a) → P'_c
Mirror(P, c) → P'_b
Mirror(P, b) → P'_a
Line(P'_c, P'_b) → i
OrthogonalLine(A, b) → j
OrthogonalLine(B, c) → k
Intersect(j, k) → H

AI Drawing Prompt Reference

Draw a triangle ABC, then select a point P on the circumcircle of the triangle. Construct perpendicular lines from P to each side of the triangle, finding their intersections with the sides as P_a, P_b, and P_c. Connect P_a and P_b to form line h. Next, reflect each side of the triangle over P to get P'_a, P'_b, and P'_c. Connect P'_a and P'_b to form line i. Finally, construct the perpendiculars from A to BC and from B to AC, and find their intersection H.

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