Given: δabc prove: the medians of δabc are concurrent. proof: statements reasons 1. the vertices of δabc are unique points: a(x1,y1), b(x2,y2), and c(x3,y3). given 2. use rigid transformations to transform δabc into δa'b'c', so that vertex a' is at the origin and a'c' lies on the x-axis in the positive direction. in the coordinate plane, any point can be moved to any other point using rigid transformations and any line can be moved to any other line using rigid transformations. 3. any property that is true for δa'b'c' will also be true for δabc. definition of congruence 4. let r, s, and t be real numbers such that the vertices of δa'b'c' are a'(0,0), b'(2r,2s), and c'(2t,0). defining constants 5. let d', e', and f' be the midpoints of a'b', b'c', and a'c' respectively. defining points 6. d' = (r , s) e' = (r + t, s) f' = (t, 0) definition of midpoints 7. slopes of lines: definition of slope 8. equations of lines: using point-slope formula 9. lines a'e' and b'f' intersect at point p. algebra 10. ? 11. all three lines contain point p. algebra 12. the three medians are concurrent. definition of concurrent 27 what is step 10 in this proof? a. statement: point p lies on line c'd'. reason: the coordinates of p satisfy the equation of line c'd'. b. statement: all three lines share point p. reason: definition of midpoint c. statement: point p lies on line a'e' and line a'e'. reason: algebra d. statement: line a'e' and line a'e' are concurrent. reason: definition of concurrence e. statement: two of the three medians share point p. reason: using point-slope formula