La circunferencia inscrita del triángulo $ABC$ se encuentra con los lados $AB, AC$ y $BC$ en $M,N$ y $P$ , respectivamente. Pruebe que el ortocentro del triángulo $MNP,$ el incentro y el circuncentro del triángulo $ABC$ son colineales. [asy]\nimport graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffwwww = rgb(1,0.4,0.4); pen xdxdff = rgb(0.49,0.49,1);\ndraw((8,17.58)--(2.84,9.26)--(20.44,9.21)--cycle); draw((8,17.58)--(2.84,9.26),ttttff+linewidth(2pt)); draw((2.84,9.26)--(20.44,9.21),ttttff+linewidth(2pt)); draw((20.44,9.21)--(8,17.58),ttttff+linewidth(2pt)); draw(circle((9.04,12.66),3.43),blue+linewidth(1.2pt)+linetype('8pt 8pt')); draw((6.04,14.42)--(8.94,9.24),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(11.12,15.48),ffwwww+linewidth(1.2pt)); draw((11.12,15.48)--(6.04,14.42),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(7.81,14.79)); draw((11.12,15.48)--(6.95,12.79)); draw((6.04,14.42)--(10.12,12.6));\ndot((8,17.58),ds); label('$A$', (8.11,18.05),NE*lsf); dot((2.84,9.26),ds); label('$B$', (2.11,8.85), NE*lsf); dot((20.44,9.21),ds); label('$C$', (20.56,8.52), NE*lsf); dot((9.04,12.66),ds); label('$O$', (8.94,12.13), NE*lsf); dot((6.04,14.42),ds); label('$M$', (5.32,14.52), NE*lsf); dot((11.12,15.48),ds); label('$N$', (11.4,15.9), NE*lsf); dot((8.94,9.24),ds); label('$P$', (8.91,8.58), NE*lsf); dot((7.81,14.79),ds); label('$D$', (7.81,15.14),NE*lsf); dot((6.95,12.79),ds); label('$F$', (6.64,12.07),NE*lsf); dot((10.12,12.6),ds); label('$G$', (10.41,12.35),NE*lsf); dot((8.07,13.52),ds); label('$H$', (8.11,13.88),NE*lsf); clip((-0.68,-0.96)--(-0.68,25.47)--(30.71,25.47)--(30.71,-0.96)--cycle);\n[/asy]