Summary — Unit I: The Four Tritangent Circle Centers (§§ 1–18)¶
Key Invariants of the Tritangent System \((S, S', S'', S''')\)
Locus |
Identity |
Geometric Meaning |
|---|---|---|
§ 1 |
\(S = \operatorname{orthocenter}(\triangle S'S''S''')\) |
\(\triangle ABC\) is the orthic (pedal) triangle of the excentral triangle \(\triangle S'S''S'''\) |
§ 2 |
\(rr'r''r''' = \Delta^2\) |
L'Huilier’s symmetric area-radius theorem (1810) |
§§ 3–4 |
\(r'r'' + r'r''' + r''r''' = s^2\), \(\sum \frac{1}{r^{(k)}} = \frac{1}{r}\) |
Elementary symmetric polynomials of the three exradii |
§ 5 |
\(r' + r'' + r''' - r = 4R\) |
Sum of signed tritangent radii equals four circumradii |
§ 6 |
\(\sum_{i < j} r_i r_j = ab + ac + bc\) |
Note that \(r\) enters with negative sign in \((-r) + r' + r'' + r''' = 4R\) |
§§ 8–9 |
\(\dfrac{\delta}{r} = \dfrac{\delta^{(k)}}{r^{(k)}} = \dfrac{\Delta}{2R}\), \(\delta' + \delta'' + \delta''' = \delta + 2\Delta\) |
Contact triangle areas scale linearly with the four tritangent radii |
§§ 16–17 |
\(SS'^2 + S''S'''^2 = 16R^2\), \(\sum_{i<j} S_i S_j^2 = 48R^2\) |
Opposite edges of the orthocentric quadrangle \(SS'S''S'''\) sum in quadrance to \((4R)^2\) |