Summary — Unit IV: Proportional Derived Triangles (§§ 61–87)

Homothetic Hierarchies of Contact and Orthic Triangles

1. Geometric-Mean Proportionality (§§ 61, 63): Both the orthic-contact operation and the contact-orthic operation square the scale factor \(\frac{\rho}{R}\) or \(\frac{r^{(k)}}{2R}\), yielding exact area geometric means:

   \[\triangle def : \triangle MNP = \triangle MNP : \Delta, \qquad \mu^{(k)} : \delta^{(k)} = \delta^{(k)} : \Delta,\]

   while rotating side orientations back into exact parallelism (homothety) with \(\triangle ABC\)!

2. Five-Triangle Area Partition (§ 64): Combining \(\mu^{(k)} = \frac{r^{(k)2}}{4R^2}\Delta\) with \(\sum r_i^2 = 4R(2R - \rho)\) yields the remarkable metric conservation law

   \[\mu + \mu' + \mu'' + \mu''' \pm \triangle MNP = 2\Delta.\]

3. Half-Perimeter & Corner Inradii (§§ 65, 71–73): The orthic triangle \(\triangle mnp\) of the contact triangle \(\triangle DEF\) has perimeter equal to one-half the perimeter of \(\triangle MNP\) (\(\frac{\Delta}{R}\)), and the corner orthic inradii \(\rho' = r\cos\alpha\), \(\rho'' = r\cos\beta\), \(\rho''' = r\cos\gamma\) satisfy \(OM + ON + OP = r + \rho + \rho' + \rho'' + \rho'''\).

4. Translation Congruence & Orthocenter S (§§ 85–87): The corner-incenter triangle \(\triangle O'O''O'''\) is congruent and parallel-sided to \(\triangle DEF\), carrying the orthocenter \(J\) of \(\triangle DEF\) into the incenter \(S = X(1)\) of \(\triangle ABC\).