Problem five states, "Given three points A, B, and C. Construct a line intersecting AC in the point X and BC in the point Y such that AX = XY = YB. Prove your construction is valid."
1. First, let's consider 3 points A, B, C. Then construct a segment from A to B and a segment from B to C. 2. Now, we will pick any point X on our segment AB. 3. Construct a circle with X as its center and radius of AX. 4. Now, construct a point Y at the intersection of our circle and of line segment BC. We can see that XY is also a radius of our circle. 5. Since AX is a radius of our circle and XY is a radius of our circle, then it follows that AX=XY. 6. Next, construct a line that is parallel to segment BC and let the intersection of this line and our circle be X'. 7. Now we have that XX' is also a radius of our circle, so AX=XY=XX'. 8. Next, construct a line that is parallel to XY and that passes through X'. 9. Construct a circle with center at X' and radius of XX'. Let Y' be the intersection of this parallel line and circle. 10. Since XX' and X'Y' are both radii of the circle centered at X' and XX'=XY=AX, then XX'=XY=AX=X'Y'. 11. Now construct a circle with center at Y' and radius of X'Y'. 12. We can see since XX' and X'Y' are both radii of our circle that XX'=X'Y'. We have already shown that AX=XY=XX', therefore AX=XY=XX'=X'Y'. 13. Construct one more circle with Y' as its center and its radius being X'Y'. 14. Now, we can see that AX = XY = YB if Y'=C. We must move our X until Y'=C. 15. Once X is moved until Y'=C, then we have finished our construction and have AX = XY = YB.
