Similarly,
∠
F
C
F
B
H
2245
∠
H F
B
F
A
and
∠
F
B
F
C
H
2245
∠
H F
C
F
A
.
Therefore,
segment
F
A
H
is the angle bisector of
∠
F
C
F
A
F
B
,
segment
F
B
H
is the angle bisector of
∠
F
C
F
B
F
A
,
and
segment
F
C
H
is the angle bisector of
∠
F
B
F
C
F
A
.
Thus,
H
is the point of concurrence of the angle bisectors of the interior
angles of
F
A
F
B
F
C
.
Thus,
H is the Incenter of
F
A
F
B
F
C
.
Q E D
Method 1:
By the Common Hypotenuse Theorem applied twice,
circle c
1
=
C(diam = AC
)
contains A, C,
F
A
, and F
C
,
and
circle c
2
=
C(diam = HC
)
contains A, H,
F
A
, and F
B
.
See the figure.
Note that
∠
F
C
F
A
H
=
∠
F
C
F
A
A
, so
∠
F
C
F
A
H
2245
∠
F
C
F
A
A
.
∠
F
C
F
A
A
and
∠
F
C
C A
are both inscribed in circle c
1
and they intercept the same arc
(F
C
A
) , so by a corollary to the Inscribed
Angle theorem ,
∠
F
C
F
A
A
2245
∠
F
C
C A .
Note that
∠
F
C
C A
=
∠
H C F
B
,
so
∠
F
C
C A
2245
∠
H C F
B
,
Now,
∠
H C F
B
and
∠
H F
A
F
B
are both inscribed in circle c
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Shirley
 Pythagorean Theorem, Hypotenuse, Inscribed angle, FB FC FA

Click to edit the document details