MathDB

Let

A_1, B_1, C_1

, be the feet of the altitudes of

\vartriangle ABC

drawn from the vertices

A, B, C

respectively, and let

M

be the orthocenter (point of intersection of altitudes) of

\vartriangle ABC

. Assume that the orthic triangle (i.e. the triangle whose vertices are the feet of the altitudes of the original triangle)

A_1

B_1

C_1

exists. Prove that each of the points

M

A

B

, and

C

is the center of a circle tangent to all three sides (extended if necessary) of

\vartriangle A_1B_1C_1

. What is the difference in the behavior of acute and obtuse triangles

ABC

3