Given any arbitrary triangle, dissect it into four triangles of equal area, using in a pattern like that shown here. Three of them share a side with the given triangle, and the remaining one does not touch any of the given sides.

This must be a compass and straightedge construction. Please do not offer answers with measurements or approximation algorithms. Relevance

Given triangle of sides a, b, c, we mark off points where the ratio of the short segments to the length of the sides are in the ratio of (1/2)(1 - 1/√5) = 0.276393... approximately. Then drawing lines from those points to the opposite vertex defines the interior triangle of area = 1/4 of original triangle. This ratio can be found by straightedge and compass, and I'll get to that later, as to a quick way to do it. Graphic to follow.

Edit: Let X, Y, Z be the vertices of a given triangle, and XY is the base. Draw line AB ⊥ XY passing through X, and mark a point C on XB such that AX = XC = 2XY, and CB = XY. Draw lines YA, YB. Then draw line DC II YB where D is on line YA, Mark E on line XY such that YE = YD. Find point F between EX such that EF = FX = (1/2)EX.

Next, mark point G on XY such that GY = XF. Draw line GH II YZ where H is on XZ, and draw line FI II YZ, where I is on line YZ. Mark point J on YZ such that JY = ZI. Then draw lines XJ, YH, ZF, which completes the drawing, after some erasing to make it look like the original diagram.

Graphic to come a bit later.

Edit 2: Here's the graphic:

http://i254.photobucket.com/albums/hh120/Scythian1...