Thales theorem objectively correct proof.

by qwerty123456asdfgzxcvb, Nov 3, 2024, 5:32 AM

let $I,J$ be the two circle points

we are trying to prove that for a fixed point $P$ on a conic through $I,J$, that for a moving point $A$ on the conic and $B$ on the conic such that \[(PA,PB;PI, PJ) = -1\], that $AB$ goes through a fixed point.

claim: this fixed point is the pole of $IJ$. let $X$ be the pole of $IJ$ and let $B'$ be the second intersection of $AX$ and the conic.

note that $ABIJ$ is harmonic on the conic, but we also have that \[I(J,X;A,B')=-1$, so $IB=IB'$ and $B=B'$, so $A,B,X$ are always collinear.

since the center of a circle is the pole of the line at infinity = $IJ$, we are done.

(this generalizes to prove that on any conic with fixed point $P$ and two points $A,B$ such that $\angle APB = 90^\circ$, $AB$ passes through a fixed point)

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