Summary — Unit II: Orthocenter, Orthic Triangle & Inradius ρ (§§ 19–48)¶
The Orthic Invariant \(\rho = 2R\cos\alpha\cos\beta\cos\gamma\) as Third Fundamental Radius
Whereas Euler (1765) expressed triangle-center invariants through \((p, q, r)\) and modern texts typically choose \((R, r, s)\), Feuerbach introduces in § 24 the signed orthic inradius \(\rho = 2R\cos\alpha\cos\beta\cos\gamma\), unlocking clean radius formulas free of \(s^2\):
Locus |
Identity |
Significance |
|---|---|---|
§ 19 |
\(MN + MP + NP = \dfrac{2\Delta}{R}\) |
Perimeter of Fagnano’s minimal-perimeter orthic triangle |
§§ 23–24 |
\(\rho = 2R\cos\alpha\cos\beta\cos\gamma\), \(\dfrac{\triangle MNP}{\Delta} = \dfrac{\rho}{R}\) |
\((A, B, C, O)\) are the four tritangent centers of \(\triangle MNP\) |
§ 26 |
\(R_{MNP} = \dfrac{1}{2}R\) |
Circumradius of \(\triangle MNP\) equals half the circumradius of \(\triangle ABC\) |
§§ 27–28 |
\(a^2 + b^2 + c^2 = 8R^2 + 4\rho R\), \(\sum r_i^2 = 4R(2R - \rho)\) |
Eliminates \(s^2\) in favor of \((R, \rho)\) |
§ 33 |
\(R(BCO) = R(ACO) = R(ABO) = R\) |
Equal circumradii of the three orthocentric sub-triangles |
§ 35 |
\(AO \cdot OM = BO \cdot ON = CO \cdot OP = 2\rho R\) |
Power of \(O\) with respect to the nine-point and circumcircles |
§ 47 |
\(Ka = \tfrac{1}{2}AO\), \(Kb = \tfrac{1}{2}BO\), \(Kc = \tfrac{1}{2}CO\) |
Altitude vertex-to-orthocenter segment is twice the circumcenter-to-side distance |