Line Segments through the Vertices and the Circumcenter of an Acute Triangle

AD

≈

9.85

BE

≈

9.25

CF

≈

11.08

R

≈

6.67

1

AD

+

1

BE

+

1

CF

≈

0.30

2

R

≈

0.30

Let ABC be an acute triangle, O its circumcenter, and R its circumradius. Let D, E, and F be the points along AO, BO, and CO across from A, B, and C. Then

1

AD

+

1

BE

+

1

CF

=

2

R

.

Details

See the First Iberoamerican Olympiad (problem 6).

External Links

Acute Triangle (Wolfram MathWorld)

Circumcenter (Wolfram MathWorld)

Circumcircle (Wolfram MathWorld)

Permanent Citation

Jay Warendorff

"Line Segments through the Vertices and the Circumcenter of an Acute Triangle"
http://demonstrations.wolfram.com/LineSegmentsThroughTheVerticesAndTheCircumcenterOfAnAcuteTri/
Wolfram Demonstrations Project
Published: March 7, 2011