Mathemarium

Hello everyone. Today we will prove theorem about central line $X_5X_6$ of triangle. Proof of the Central Line in a Golden Rectangle Construction Statement of the Theorem Let $ABCD$ be a golden rectangle where $\frac{AB}{BC} = \phi$, and construct the square $BCQP$ inside it. Reflect $P$ over $D$ to obtain $E$. Then, the line $EB$ coincides with the central line $X_5X_6$ of triangle $ABP$. Step-by-Step Proof 1. Define the Coordinates We assign coordinates as follows: $A = (0,0)$, $B = (\phi x, 0)$, $C = (\phi x, x)$, $D = (0, x)$. The square $BCPQ$ ensures $P = (\phi x, 2x)$. The reflection of $P$ across $D$ is $E = (-\phi x, 2x)$. 2. Compute the Nine-Point Center $X_5$ The nine-point center $X_5$ is the circumcenter of the medial triangle, which consists of the midpoints: $$M_1 = \left(\frac{0 + \phi x}{2}, 0\right) = \left(\frac{\phi x}{2}, 0\right),$$ \[ M_2 = \left(\f...