Pope
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Pope asked in Science & MathematicsMathematics · 7 years ago

Construct a triangle quadrisection?

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.

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  • 7 years ago
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    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...

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