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Let the circle

\Omega

and the line

\ell

intersect at two different points

A{}

and

B{}

. For different and non-points. Let

X

and

T

be points on

\ell

and

Y

and

Z

be points on

\Omega

, all of them different from

A{}

and

B{}

. Prove the following statements:
[*]The points

X,Y

and

Z

lie on the same line if and only if

\frac{\overline{AX}}{\overline{BX}}=\pm\frac{AY}{BY}\cdot\frac{AZ}{BZ}.

[*]The points

X,Y,Z

and

T

lie on the same circle if and only if

\frac{\overline{AX}}{\overline{BX}}\cdot\frac{\overline{AT}}{\overline{BT}}=\pm\frac{AY}{BY}\cdot\frac{AZ}{BZ}.

Note: In both points, the sign

+

is selected in the right parts of the equalities if the points

Y{}

and

Z{}

lie on the same arc

AB

of the circle

\Omega

, and the sign

-

Y{}

and

Z{}

lie on different arcs

AB

. By

\overline{AX}/\overline{BX}

, we indicate the ratio of the lengths of

AX

and

BX

, taken with the sign

+

-

depending on whether the

AX

and

BX

vectors are co-directed or oppositely directed.
Proposed by M. Skopenkov

Problem Statement