Problem: From a vertex of a certain triangle, draw the altitude, the angle bisector, and the median. Being given that these three lines divide the angle  at the vertex into four equal parts, find the size of the angle at the vertex.

Solution:

 

Consider two right triangles formed by the altitude:

Using A, B, and C for the measures of the angles at these respective vertices,   the two angles at vertex   A   are   and  .   Then      and    .

Consider the two triangles formed by the median:

The division of the angles at the vertex  A  gives

in the acute triangle and

in the obtuse triangle.

The Law of Sines gives:

ACUTE Triangle:

OBTUSE Triangle:

Therefore:

and since

                            

we have

Since sin 2ø = sin ø cos ø, we have

So