1. Draw a triangle ABC on your paper. Choose a point P inside the triangle. Call the barycentric coordinates of P (x,y,z), where A=(0,1,0), B=(0,0,1) and C=(1,0,0). Construct a line m through A and a n line parallel to side BC such that the intersection of the lines m and n is at P.
a. Using what you learned in lab, in terms of the barycentric coordinates of P find the equations defining m and n.
b. What are ratios of
the areas of triangles: ABP/ABC, APC/ABC, and PBC/ABC?
2. The following set of points lie on only three distinct lines. Sort them into which points are on which lines (note that some of the points are the intersections of the lines), and give equations in terms of barycentric coordinates (x,y,z) for the lines.
A=(0,0,1), B=(1/3,0,2/3), C=(1/7,4/7,2/7), D=(2/3,-1,4/3),E=(2,0,-1),
F=(0,2/3,1/3), G=(1/4,1/2,1/4), H=(1/5,2/5,2/5)
3. Draw a triangle ABC on your paper, and a point P with coordinates (x,y,z) inside ABC. Draw the line AP and label the intersection with BC by X.
a. What is the ratio of the areas of the triangles ABX/ACX?
b. What is the ratio of the areas of the triangles PBX/PCX?
c. Use the equations from a. and b. to find the ratio of the areas of the triangles APB/APC.
4. Draw a triangle ABC on your paper, and construct the angle bisector at vertex A. Label the lengths of the sides opposite vertices A, B, and C by a, b, and c, respectively. Label the point where the bisector meets side BC by X.
a. What are the barycentric coordinates of X in terms of a, b, and c? What is the equation of the bisector AX?
b. Repeat the calculation
for vertex B's angle bisector. What
are the barycentric coordinates of the incenter of triangle ABC?
5. Draw a triangle ABC on your paper (not too small). Divide the sides of the triangle into five equal parts. Now connect the corresponding points on pairs of sides of the triangles to form a grid such as the ones that you made out of congruent copies of one small triangle.
a. Find barycentric coordinates for five of the grid points.
b. Find two lines connecting grid points on the edges of triangle ABC that do not pass through any of the interior grid points and two that do pass through interior grid points (without being parallel to the sides of the triangle). What are the coordinates of the intersections of your lines?
c. If your extend your grid outside triangle ABC will the lines still avoid meeting more grid points?