We start with the original triangle:
Now what we are going to do is prove that
A = FEC.
We are going to do this by looking at quadrilateral OFCE.
Since this quadrilateral is inscribed
in the triangle and formed by the perpendiculars, we know that
F and E,
are right angles.
Additionally looking back, if we connect
EF and OC, we see that FCO and FEO
(in orange) both share arc OF.
Therefore, FCO
= FEO (ACD
= FEA). Now look at triangle ACD.
We know that triangle ACD is a right triangle,
since D is one of the perpendiculars and therefore a right angle.
Due to this fact we know that A and
C must therefore add to 90 degrees
(A and C
are complementary).
Now look at the figure below:
We additionally can see that since E forms a right angle, we know that CEF must be complementary with FEA.
Now since above we showed that ACD
(FCO) was equal to FEA
(FEO), we must conclude that FEC = A. This
is the second part of what we needed to show.
