perplexus.info :: Geometry : Lattice Point as Centroid

Lattice Point as Centroid (Posted on 2011-04-10)

The three vertices of a triangle are lattice points. The triangle contains no other lattice points but its interior contains exactly one lattice point.

Prove that the interior lattice point is the triangle's centroid.

Submitted by Bractals

Rating: 3.0000 (1 votes)

Solution: (Hide)

Let the three vertices be A, B, and C and P the interior lattice point. By Pick's Theorem we have
Area(ΔPAB) = Area(ΔPAB) = Area(ΔPAB) = I + B/2 - 1 = 0 + 3/2 - 1 = 1/2
The centroid is the only interior point P of ΔABC having the property that
Area(ΔPAB) = Area(ΔPAB) = Area(ΔPAB)

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(3): One Solution (spoiler)	Dej Mar	2011-04-11 16:17:55
Typo in solution	Bractals	2011-04-11 14:03:20
re(4): One Solution (spoiler)	Steve Herman	2011-04-11 13:19:51
re(3): One Solution (spoiler)	Steve Herman	2011-04-11 13:17:39
re(2): One Solution (spoiler)	Jer	2011-04-11 11:40:09
re(2): One Solution (spoiler)	Steve Herman	2011-04-11 09:49:49
re: One Solution (spoiler)	Dej Mar	2011-04-11 05:26:38
One Solution (spoiler)	Steve Herman	2011-04-11 01:25:53

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (10)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info