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A Practical Use for the Orthocenter

Date: 03/07/2001 at 17:35:48
From: Dustin Hodel
Subject: Orthocentre

My geometry teacher proposed a question, or challenge really, to find 
a practical use for the orthocentre. I've looked all over for an 
answer, with no luck. Please help.

Date: 03/09/2001 at 07:35:44
From: Doctor Floor
Subject: Re: Orthocentre

Hi, Dustin,

Thanks for writing.

It seems very difficult to find a "practical" use for the orthocenter. 
But perhaps the following story will help you.

Jonathan is competing in a bicycle cross-country race. The track will 
be a triangular one (with acute angles). Jonathan's mom wants to see 
her son ride the bike on each straight part of the track, and she 
wants to walk the minimum distance possible.

So she connects each "vertex" of the course with its orthocenter, and 
sees where each line meets the opposite part of the track. This way 
she finds the three points where she can see her son ride the bicycle, 
while having to walk the minimum total distance.

The property used here is this: The triangle formed by the feet of the 
altitudes of ABC is the triangle inscribed in ABC with the smallest 

I hope this helps!

Best regards,
- Doctor Floor, The Math Forum   
Associated Topics:
High School Geometry
High School Practical Geometry
High School Puzzles
High School Triangles and Other Polygons

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