Triangle Divided (Posted on 2010-04-15)

Let P be a point in the interior of an equilateral triangle.

Three line segments connect P with the vertices of the 
triangle and three line segments connect P perpendicularly
to the sides of the triangle.

These six line segments divide the triangle into six smaller
triangles that surround P.

If u, v, w, x, y, and z denote the areas of the triangles 
around P in that order, then prove that

                     u + w + y = v + x + z.

	Submitted by Bractals
Rating: 2.5000 (2 votes)
Solution:	(Hide)
For each side of the equilateral triangle construct a line through P that is parallel to that side. Each line segment from P to a vertice is a diagonal of a parallelogram formed by two of the parallel lines thus constructed and two sides of the original equilateral triangle. Thus, it divides it into two congruent triangles. Each line segment from P to a side is an altitude of an equilateral triangle formed by that side and two of the parallel lines so constructed. Thus, it divides it into two congruent triangles. Thus, the original equilateral triangle is divided into twelve smaller triangles. Color the triangle areas denoted by u, w, and y - leave those denoted by v, x, and z as uncolored. Six of the twelve triangles are colored and for each colored triangle, there exists a corresponding uncolored triangle that is congruent. Therefore, u + w + y = v + x + z. Note: The above explanation seems more complicated than looking at the constructed figure.

	Subject	Author	Date
	re(4): Easy when you know how	broll	2010-04-19 04:32:53
	re(3): Easy when you know how	Harry	2010-04-18 21:45:01
	re(2): Easy when you know how	broll	2010-04-18 02:54:55
	re: Easy when you know how	Harry	2010-04-17 23:11:25
	Easy when you know how	broll	2010-04-17 07:39:28
	Approach	broll	2010-04-16 14:46:46
	re: Approach.	Bractals	2010-04-16 08:12:58
	Approach.	Vee-Liem Veefessional	2010-04-16 00:13:22