Question

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and
∠BA1C+∠CB1A+∠AC1B=480∘
Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$
Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$
and $CC_1C_2$ all pass through two common points.

Answer 1

This is really hard...

Answer 2

I got you @xxaikoxx

Answer 3

Consider an equilateral triangle $ABC$ with interior points $A_1, B_1, C_1$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $\angle BA_1C + \angle CB_1A + \angle AC_1B = 480^\circ$. We're aiming to prove that if triangle $A_1B_1C_1$ is scalene, then the circumcircles of triangles $AA_1A_2$, $BB_1B_2$, and $CC_1C_2$ share two prevalent points.

Let's analyze the quandary step by step:

1. Optically canvass that $\angle A_1C_1B_1 + \angle B_1A_1C_1 + \angle C_1B_1A_1 = 360^\circ$, which implicatively insinuates that $\angle BA_2C + \angle CB_2A + \angle AC_2B = 360^\circ$, making points $A_2, B_2, C_2$ collinear. This is due to the fact that the exterior angles of a triangle integrate up to $360^\circ$.

2. Note that the configuration suggests an obnubilated equilateral triangle $A_2B_2C_2$ within the more sizably voluminous triangle $ABC$. This obnubilated equilateral triangle is composed by the cevians $BA_1$, $CB_1$, and $AC_1$.

3. Considering this obnubilated equilateral triangle $A_2B

Answer 4

Yk the dollar sign is latex on whatever respective site Aiko is on ;-;