Points $A_1,A_2,A_3,A_4$ are not concyclic, the same for points $B_1,B_2,B_3,B_4$. For all $i, j, k$ the circumradii of triangles $A_iA_jA_k$ and $B_iB_jB_k$ are equal. Can we assert that $A_iA_j=B_iB_j$ for all $i, j$'?
Problem
Source: 2021 Sharygin Geometry Olympiad Finals grade VIII p5
Tags: geometry, equal circles, circumradius, equal segments
JustinLee2017
12.06.2022 21:38
No. [asy][asy] import graph; size(15cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.0510864745011075, xmax = 14.312549889135251, ymin = -6.9096230598669575, ymax = 6.64248337028825; /* image dimensions */ /* draw figures */ draw((-1,2)--(6,2), linewidth(1)); draw((6,2)--(6,-1), linewidth(1)); draw((6,-1)--(-1,-1), linewidth(1)); draw((-1,-1)--(-1,2), linewidth(1)); draw((-1,-1)--(-1.9866075388026592,0.5), linewidth(1) + red); draw((-1,2)--(6.98660753880266,0.5), linewidth(1) + red); draw((-1.9866075388026592,0.5)--(6.98660753880266,0.5), linewidth(1) + linetype("2 2")); /* dots and labels */ dot((-1,-1),dotstyle); label("$A_1$", (-0.9247006651884698,-0.8253658536585329), NE * labelscalefactor); dot((6,-1),dotstyle); label("$B_2$", (6.064212860310421,-0.8253658536585329), NE * labelscalefactor); dot((6,2),dotstyle); label("$A_2$", (6.064212860310421,2.1724168514412447), NE * labelscalefactor); dot((-1,2),dotstyle); label("$B_1$", (-0.9247006651884698,2.1724168514412447), NE * labelscalefactor); dot((2.5,0.5),linewidth(4pt) + dotstyle); label("$E$", (2.5697560975609752,0.6469179600886952), NE * labelscalefactor); dot((6.98660753880266,0.5),dotstyle); label("$A_4 = B_3$", (6.578625277161861,1.1613303769401364), NE * labelscalefactor); dot((-1.9866075388026592,0.5),linewidth(4pt) + dotstyle); label("$A_3 = B_4$", (-2.982350332594234,1.072638580931267), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]