Lab 2

Part 1 – Computing Barycentric Coordinates

Construct the triangle ABC and any point D.  Then construct the 3 lines through D parallel to the sides of ABC.  From this calculate 3 ratios that are the barycentric coordinates x, y, z for D (the sum x+y+z should equal 1 always, even if D is outside the triangle.

Use this tool to investigate some properties of barycentric coordinates.  It may help to sometimes Merge the Point D with another point or an object.

Case 1.  What are the barycentric coordinates of the various points of the median through A of the triangle ABC?  Does this depend on the triangle?

Case 2.  What are the barycentric coordinates of the centroid of the triangle ABC?  Does this depend on the triangle?

Case 3.  What are the barycentric coordinates of the incenter of the triangle ABC?

Compare these 3 numbers with the lengths of each side, divided by the perimeter.  What do you find?  Can you explain this in the light of what you know about angle bisectors and also Ceva's theorem?

Case 4.  Construct the incircle of the triangle ABC.  If the points of tangency of the circle are A' on BC, B' on CA, C' on AB, find the barycentric coordinates of the intersection of AA' and BB'.  Compare these 3 numbers with the tangent lengths BA', CB', AC'.  Explain using barycentric coordinates and/or Ceva's Theorem, why the 3 lines AA', BB', CC' are concurrent. (This is a problem on Assignment 3 also.)

Part 2- Ceva and Menelaus Experiment

Draw a triangle ABC with extended sides (LINES not segments).  Construct points A', B', C' on sides BC, CA, AB.

Compute the 3 ratios BA'/A'C, CB'/B'A, AC'/C'B and also the product of the ratios (If you get a different order, you may find your ratios are the negative of the ones above.  That is OK.  Just keep is consistent and see what the product is.) 

Ceva

Construct the lines AA' BB', CC'.  Drag the points A', B', C' to positions so that the lines appear to be concurrent.  What is the product of the ratios from above (approximately)?

Menelaus

Now construct lines A'B', B'C', C'A'. Drag the points A', B', C' to positions so that the lines appear to be the same (i.e., the points A', B', C' are collinear).  What is the product of the ratios from above (approximately)?

Part 3 - Constructing an Affine Transformation

• Draw a triangle ABC.  Also draw a point D.
• Measure the abscissa x and ordinate y of D. 
• Mark x as ratio and dilate A with center C by ratio x to get A'
• Mark y as ratio and dilate B with center C by ratio y to get B'.
• Construct D' as either the translation of A' by vector CB' or as the translation of B' by vector CA'.  The figure CA'D'B' should be a parallelogram. 
• If T is the affine transformation that maps (0,0) to C, (1,0) to A and (0,1) to B, then D' is the image of D.  Make a tool that takes A, B, C, D and construct D'.

Image of a Circle and its Chords

• Draw a circle.  Merge D to the circle and construct the locus by selecting D' and D.  The locus of D' is the image of the circle.
• Figure out how to construct the locus of midpoints of parallel chords of the ellipse and tell what you see. Hint:  Construct a chord or chords in the circle first and then the image in the ellipse.

You can investigate the images of other figures in a similar way.