Unit IV: Fifth Section — Comparative Proportions Among Derived Triangles (§§ 61–87)¶
Karl Wilhelm Feuerbach — Properties of Several Remarkable Points of the Rectilinear Triangle (1822, pp. 41–55, Fig. 8–12)
§§ 61–62. Mean Geometric Proportionality of ΔMNP and ΔABC¶
Let \(M, N, P\) in triangle \(ABC\) be the feet of the altitudes dropped from \(A, B, C\) upon the opposite sides, intersecting in \(O\), and in triangle \(MNP\) let \(d, e, f\) be the contact points of its inscribed circle (inradius \(\rho\)), while \(d', e', f'\) are the contact points of its excircle in the sector of \(\angle PMN\) (Fig. 8).
§ 61 (Mean-Proportional Area Law): Because (§ 26) the circumradius of \(\triangle MNP\) equals \(\tfrac{1}{2}R\), applying § 8 inside \(\triangle MNP\) gives \(\triangle def : \triangle MNP = \rho : 2(\tfrac{1}{2}R) = \rho : R\); and since (§ 25) \(\triangle MNP : \Delta = \rho : R\), it follows that:
\[\triangle def : \triangle MNP = \triangle MNP : \Delta\](and for an obtuse triangle at \(A\), \(\triangle d'e'f' : \triangle MNP = \triangle MNP : \Delta\)). Thus: The orthic triangle \(MNP\) is the mean geometric proportional triangle between \(\triangle ABC\) and the contact triangle formed by that tritangent circle of \(\triangle MNP\) whose center falls at the orthocenter \(O\) of \(\triangle ABC\). Moreover, because \(ON \perp AC\) and \(ON \perp df\), \(AC \parallel df\) and likewise \(AB \parallel de\), \(BC \parallel ef\): triangles \(ABC\) and \(def\) are homothetic with parallel corresponding sides!
By dual reasoning (since \(\triangle ABC\) is the orthic triangle of \(\triangle S'S''S'''\)), every triangle is itself the mean geometric proportional triangle between the contact triangle of any one of its four tritangent circles and the triangle formed by the centers of the remaining three.
§ 62 (Perimeter Ratio): Since \(de = 2\rho\sin\gamma\), \(df = 2\rho\sin\beta\), \(ef = 2\rho\sin\alpha\) and \(\sin\alpha + \sin\beta + \sin\gamma = \frac{\Delta}{rR}\), comparison with (§ 19) \(MN + MP + NP = \frac{2\Delta}{R}\) yields:
\[(de + df + ef) : (MN + MP + NP) = \frac{2\rho\Delta}{rR} : \frac{2\Delta}{R} = \rho : r.\]
§§ 63–65. Orthic Triangles μ, μ’, μ’’, μ’’’ Inside the Four Contact Triangles¶
§ 63 (Mean-Proportional Chain): Let \(D, E, F\) be the contact points of the incircle of \(\triangle ABC\), and let \(m, n, p\) be the feet of the altitudes of \(\triangle DEF\) intersecting at its orthocenter \(J\) (Fig. 9). Because \(\angle FDE = 90^\circ - \tfrac{1}{2}\alpha\), \(\angle DEF = 90^\circ - \tfrac{1}{2}\beta\), \(\angle EFD = 90^\circ - \tfrac{1}{2}\gamma\), § 23 gives \(\triangle mnp = 2\,\triangle DEF\,\sin\tfrac{1}{2}\alpha\sin\tfrac{1}{2}\beta\sin\tfrac{1}{2}\gamma = \frac{r}{2R}\triangle DEF\), while (§ 8) \(\frac{\triangle DEF}{\Delta} = \frac{r}{2R}\). Denoting by \(\delta, \delta', \delta'', \delta'''\) the four contact triangles and by \(\mu, \mu', \mu'', \mu'''\) their respective orthic triangles, setting \(a, b, c\) negative successively yields the fourfold geometric-mean chain:
\[\mu : \delta = \delta : \Delta, \qquad \mu' : \delta' = \delta' : \Delta, \qquad \mu'' : \delta'' = \delta'' : \Delta, \qquad \mu''' : \delta''' = \delta''' : \Delta,\]and all five triangles \(\mu, \mu', \mu'', \mu''', ABC\) are mutually similar with parallel corresponding sides!
§ 64 (The Five-Triangle Area Sum = 2Δ, p. 44): Since \(\mu^{(k)} = \frac{r^{(k)2}}{4R^2}\Delta\), summing via (§ 28) \(r^2 + r'^2 + r''^2 + r'''^2 = 4R(2R - \rho)\) and (§ 25) \(\frac{\rho\Delta}{R} = \triangle MNP\) yields for any acute triangle:
\[\mu + \mu' + \mu'' + \mu''' + \triangle MNP = 2\Delta\](and for an obtuse triangle, \(\mu + \mu' + \mu'' + \mu''' - \triangle MNP = 2\Delta\)).
§ 65 (Half-Perimeter Theorem): Since \(mn = r\sin\gamma\), \(mp = r\sin\beta\), \(np = r\sin\alpha\) and \(\sin\alpha + \sin\beta + \sin\gamma = \frac{s}{2R} = \frac{\Delta}{rR}\), comparison with \(MN + MP + NP = \frac{2\Delta}{R}\) gives:
\[mn + mp + np = \frac{1}{2}(MN + MP + NP).\]
§§ 71–87. Corner Inradii ρ’, ρ’’, ρ’’’ and the Orthocenter of ΔO’O’’O’’’¶
§§ 71–73 (Corner Orthic Inradii, p. 48): Denoting by \(\mathfrak{r} = \frac{r^2}{2R}\) the inradius of \(\triangle mnp\) (§ 66), and inscribing circles of centers \(O', O'', O'''\) and radii \(\rho', \rho'', \rho'''\) in the three corner triangles \(\triangle ANP, \triangle BMP, \triangle CMN\) (Fig. 11), similarity (\(\triangle ANP \sim \triangle ABC\) with ratio \(\cos\alpha\)) gives \(\rho' = r\cos\alpha\), \(\rho'' = r\cos\beta\), \(\rho''' = r\cos\gamma\). Since \(\cos\alpha + \cos\beta + \cos\gamma = \frac{r + R}{R}\) (§ 32):
\[\begin{split}OM + ON + OP &= \rho + 2(r + \mathfrak{r}), \\ \rho' + \rho'' + \rho''' &= r + 2\mathfrak{r}, \\ OM + ON + OP &= r + \rho + \rho' + \rho'' + \rho'''.\end{split}\]
§§ 85–87 (Congruence of ΔO’O’’O’’’ with ΔDEF & Incenter as Orthocenter, p. 55): The corner incenters \(O', O'', O'''\) of \(\triangle ANP, \triangle BMP, \triangle CMN\) are simultaneously the orthocenters of the corner contact triangles \(\triangle AEF, \triangle BDF, \triangle CDE\), and \(\triangle O'O''O'''\) is congruent to \(\triangle DEF\) with parallel sides (\(O'O'' \parallel DE\), \(O'O''' \parallel DF\), \(O''O''' \parallel EF\)). In § 87 (the closing analytical theorem of the monograph), Feuerbach computes directly:
\[O'S = 2r\sin\tfrac{1}{2}\alpha = DJ, \qquad O''S = 2r\sin\tfrac{1}{2}\beta = EJ, \qquad O'''S = 2r\sin\tfrac{1}{2}\gamma = FJ,\]concluding: “The center \(S\) of the circle inscribed in triangle \(ABC\) is simultaneously the intersection point of the altitudes in triangle \(O'O''O'''\) .”