### Theorem Any 2 lines $\ell_1$ and $\ell_2$ that intersect a circle at (not necessarily unique) points $A,B$ and $C,D$ respectively will have an intersection point $X$ with the property that the product of the lengths of the segments $m\overline{AX}\cdot m \overline{BX}$ and $m\overline{CX}\cdot m \overline{DX}$ will be the same. It follows that other lines that pass through this intersection point will be split in the same way. This will be true in each of the following diagrams. The distances from $A,B,C,D$ to $x$ will be $a,b,c,d$ respectively, so for this page, $a\cdot b = c\cdot d$ This is the result of the AA similarity $\triangle AXC \sim \triangle DXB$ which provides the equivalent scale factors $\frac{a}{d}=\frac{c}{b}$, all of which depends on the congruences: - $\angle XAC \cong \angle XDB$ (magenta), - $\angle XCA \cong \angle XBD$ (cyan), and/or - $\angle AXC \cong \angle DXB$ (green) #### Chord Intersection Theorem ![[int-ch-ch-thm.svg]] #### Secant Intersection Theorem ![[int-sec-sec-thm.svg]] #### Secant-Tangent Intersection Theorem ![[int-sec-tan-thm.svg]] #### Converse: Test Concyclicity If 4 points $A,B,C,D$ have lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ intersecting at point $P$ between both of or neither of (but not one of) the pairs $A,B$ and $C,D$, the concyclicity of $A,B,C,D$ is equivalent to the validity of $PA\cdot PB=PC\cdot PD $\left(P\in(\overline{AB}\cap\overline{CD})\right) \lor \left(P\in(\overleftrightarrow{AB}\cup\overleftrightarrow{CD}\setminus(\overline{AB}\cup\overline{CD})\right)$ #### Power of a point [[Power of a point with respect to a circle]]