Distance between Incenter and Fletcher Point

d	r(4R+r) 2 (4R+r) -3 2 s
2.11878	2.11878

Given a triangle

ABC

, draw the incircle. The tangent points

A

1

,

B

1

,

C

1

define the contact triangle. Let lines

AB

and

A

1

B

1

intersect at

C

2

,

AC

and

A

1

C

1

at

B

2

,

BC

and

B

1

C

1

at

A

2

. The points

A

2

,

B

2

,

C

2

are collinear and form the perspectrix of the contact triangle, also called the Gergonne line, shown in orange.

In the Soddy construction, the triangle vertices form the centers of three mutually tangent circles. There are two circles tangent to those three original circles, the inner and outer Soddy circles, with centers called the Soddy points (not labeled). The two Soddy points define the Soddy line (red), which includes the incenter

X

1

of

ABC

.

The Gergonne line and the Soddy line are perpendicular and intersect at the Fletcher point

X

1323

.

Let

R

,

r

,

s

be the circumradius, inradius and semiperimeter of

ABC

, respectively.

Define

d=

X

1

X

1323



(shown in red).

Then

d=

r(4R+r)

2

(4R+r)

-3

2

s

.

You can drag the points

A

,

B

and

C

.

External Links

Fletcher Point (Wolfram MathWorld)

Gergonne Line (Wolfram MathWorld)

Soddy Centers (Wolfram MathWorld)

Soddy Line (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan

"Distance between Incenter and Fletcher Point"
http://demonstrations.wolfram.com/DistanceBetweenIncenterAndFletcherPoint/
Wolfram Demonstrations Project
Published: July 12, 2022