Geometry Problem. GeoGebra, HTML5 Animation for iPad and Nexus.

Post your solution in the comments box below.

Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the dynamic geometry demostration of problem 889.

## Saturday, June 15, 2013

### Problem 889: Carnot's Theorem in an acute triangle, Circumcenter, Circumradius, Inradius

Labels:
acute,
Carnot theorem,
circumcenter,
circumradius,
inradius,
triangle

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OA1 + OB1 + OC1

ReplyDelete= R (cosA + cosB + cosC)

= R (1 + 4 sinA/2 sinB/2 sinC/2)

= R + 4R sinA/2 sinB/2 sinC/2

= R + r

The last step follows from

ReplyDeleter (cotB/2 + cotC/2) = a = 2R sinA = 4R sinA/2 cosA/2

r = 4R sinA/2 sinB/2 sinC/2