Minimalist Principals

By: John Vereen

            Lets construct the triangle of minimum perimeter inscribed in a given triangle. Depicted above, we have a general triangle ABC and a general inscribed triangle DEF and a segment that connects C to D. In order to construct this triangle, I reflected segment CD across both AC and CB to give me segments CD and CD. We will use these two new triangles CDD and CDD in order to construct this minimal perimeter triangle.

Above, we have a similar image to the first image depicted. However, there are a few key differences. First of all, segment CD is orthogonal to segment AB. Secondly, the segments DE, EF, and FD compose a straight line. My claim is that triangle DEF is the triangle with minimum perimeter.

Why is triangle DEF the triangle with minimum perimeter? Well, because we reflected CD and reflection is an isometry, distances and angle measures are preserved. Therefore, triangles CDD and CDD are isosceles triangles because, respectively, they have the same measures for each side and angle DCE=DCE and angle DCF=DCF.

Now lets observe the angle bisectors AC and BC for each respective isosceles triangle CDD and CDD. Because we have constructed two other segments DE and DE of equal length that share an angle bisector, then we have another isosceles triangle DDE. We have the same relationship between segments DF and DF and isosceles triangle DDF. This means that segments DE and DF are at minimal length under one condition.

Now, segments DE, DF, and EF will only be at minimal length when DE, EF, and FD compose a straight line DD. This means that segment EF covers the shortest distance possible between the two isosceles triangles, which is a straight line. The only way that DD will be a straight line is when CD is orthogonal to AB. Thus, I have proven that triangle DEF is the minimal triangle when we have the specific conditions of the construction above.