Reflection in a Circle

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2022-01-16 18:48

Circle Coordinate Geometry

This example demonstrates the reflection of a point across a circle and its geometric properties. Key math concepts include circle, reflection symmetry, and coordinate geometry. Construction highlights involve commands like Line, OrthogonalLine, Point, Midpoint, Circle, Intersect, Sequence, and Mirror in GeoGebra. Suitable for high school students learning coordinate geometry and geometric transformations.

GeoGebra Commands & Parameters

Line(A, B) → f
OrthogonalLine(A, f) → g
Point(g) → C
Line(C, f) → h
Midpoint(A, C) → D
Circle(D, C) → c
Circle(A, C) → d
Intersect(d, f) → E, F
Sequence(Translate[c, (u * (F - A))], u, -30, 30) → l1, {(x + 7.741137854794433)² + (y + 132.82651123010052)² = 5.249034179718787, (x + 7.179081177746875)² + (y + 128.27895925011595)² = 5.249034179718787, (x + 6.617024500699314)² + (y + 123.7314072701314)² = 5.249034179718787, (x + 6.054967823651753)² + (y + 119.18385529014685)² = 5.249034179718787, (x + 5.492911146604193)² + (y + 114.6363033101623)² = 5.249034179718787, (x + 4.930854469556634)² + (y + 110.08875133017774)² = 5.249034179718787, (x + 4.368797792509072)² + (y + 105.5411993501932)² = 5.249034179718787, (x + 3.806741115461511)² + (y + 100.99364737020865)² = 5.249034179718787, (x + 3.2446844384139517)² + (y + 96.44609539022409)² = 5.249034179718787, (x + 2.6826277613663922)² + (y + 91.89854341023954)² = 5.249034179718787, (x + 2.120571084318831)² + (y + 87.350991430255)² = 5.249034179718787, (x + 1.5585144072712698)² + (y + 82.80343945027043)² = 5.249034179718787, (x + 0.9964577302237103)² + (y + 78.25588747028588)² = 5.249034179718787, (x + 0.43440105317615085)² + (y + 73.70833549030132)² = 5.249034179718787, (x - 0.12765562387141038)² + (y + 69.16078351031678)² = 5.249034179718787, (x - 0.6897123009189716)² + (y + 64.61323153033223)² = 5.249034179718787, (x - 1.251768977966531)² + (y + 60.06567955034766)² = 5.249034179718787, (x - 1.8138256550140914)² + (y + 55.51812757036311)² = 5.249034179718787, (x - 2.375882332061652)² + (y + 50.97057559037856)² = 5.249034179718787, (x - 2.937939009109212)² + (y + 46.423023610394004)² = 5.249034179718787, (x - 3.4999956861567725)² + (y + 41.87547163040946)² = 5.249034179718787, (x - 4.062052363204333)² + (y + 37.3279196504249)² = 5.249034179718787, (x - 4.624109040251893)² + (y + 32.78036767044035)² = 5.249034179718787, (x - 5.1861657172994535)² + (y + 28.232815690455798)² = 5.249034179718787, (x - 5.748222394347014)² + (y + 23.685263710471247)² = 5.249034179718787, (x - 6.310279071394574)² + (y + 19.137711730486696)² = 5.249034179718787, (x - 6.872335748442135)² + (y + 14.590159750502142)² = 5.249034179718787, (x - 7.434392425489695)² + (y + 10.042607770517588)² = 5.249034179718787, (x - 7.996449102537255)² + (y + 5.495055790533036)² = 5.249034179718787, (x - 8.558505779584817)² + (y + 0.9475038105484836)² = 5.249034179718787, (x - 9.120562456632376)² + (y - 3.600048169436069)² = 5.249034179718787, (x - 9.682619133679935)² + (y - 8.147600149420622)² = 5.249034179718787, (x - 10.244675810727497)² + (y - 12.695152129405173)² = 5.249034179718787, (x - 10.806732487775058)² + (y - 17.242704109389727)² = 5.249034179718787, (x - 11.368789164822617)² + (y - 21.790256089374278)² = 5.249034179718787, (x - 11.930845841870177)² + (y - 26.337808069358832)² = 5.249034179718787, (x - 12.492902518917738)² + (y - 30.885360049343383)² = 5.249034179718787, (x - 13.0549591959653)² + (y - 35.43291202932794)² = 5.249034179718787, (x - 13.617015873012859)² + (y - 39.98046400931249)² = 5.249034179718787, (x - 14.179072550060418)² + (y - 44.528015989297046)² = 5.249034179718787, (x - 14.74112922710798)² + (y - 49.0755679692816)² = 5.249034179718787, (x - 15.30318590415554)² + (y - 53.62311994926615)² = 5.249034179718787, (x - 15.8652425812031)² + (y - 58.1706719292507)² = 5.249034179718787, (x - 16.42729925825066)² + (y - 62.718223909235256)² = 5.249034179718787, (x - 16.98935593529822)² + (y - 67.2657758892198)² = 5.249034179718787, (x - 17.55141261234578)² + (y - 71.81332786920436)² = 5.249034179718787, (x - 18.113469289393343)² + (y - 76.3608798491889)² = 5.249034179718787, (x - 18.6755259664409)² + (y - 80.90843182917345)² = 5.249034179718787, (x - 19.237582643488462)² + (y - 85.45598380915801)² = 5.249034179718787, (x - 19.799639320536023)² + (y - 90.00353578914256)² = 5.249034179718787, (x - 20.36169599758358)² + (y - 94.55108776912712)² = 5.249034179718787, (x - 20.923752674631146)² + (y - 99.09863974911167)² = 5.249034179718787, (x - 21.485809351678704)² + (y - 103.64619172909622)² = 5.249034179718787, (x - 22.04786602872626)² + (y - 108.19374370908078)² = 5.249034179718787, (x - 22.609922705773826)² + (y - 112.74129568906532)² = 5.249034179718787, (x - 23.171979382821384)² + (y - 117.28884766904987)² = 5.249034179718787, (x - 23.734036059868945)² + (y - 121.83639964903443)² = 5.249034179718787, (x - 24.296092736916506)² + (y - 126.38395162901898)² = 5.249034179718787, (x - 24.858149413964064)² + (y - 130.93150360900353)² = 5.249034179718787, (x - 25.42020609101163)² + (y - 135.47905558898807)² = 5.249034179718787, (x - 25.982262768059186)² + (y - 140.02660756897265)² = 5.249034179718787}
Circle(G, H) → e
Mirror(f, e) → f'
Mirror(h, e) → h'
Mirror(l1, e) → l1'

AI Drawing Prompt Reference

On a coordinate plane, draw a line through two given points and construct a perpendicular from a given point to that line, marking the foot of the perpendicular. Use the midpoint of the segment joining the original point and the foot as the center of a circle, and also draw another circle determined by the original point and the foot. Find the intersection points between the second circle and the original line, then use one of these intersections to generate a short family of points along a translated arc/curve around the first circle. Finally, take the first circle as the mirror and construct the reflections of the original line, the perpendicular line, and the generated curve/point sequence across that circle, producing their mirrored counterparts.

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