Kate Berryman
cavaleri@uga.edu
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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.
| First, let's consider 3 points A, B, C. Then construct a segment from A to B and a segment from B to C. |  |
| Now, we will pick any point X on our segment AB. Click here to move the X. |
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| Construct a circle with X as its center and radius of AX. |  |
| 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. |  |
| Since AX is a radius of our circle and XY is a radius of our circle, then it follows that AX=XY. |  |
| Next, construct a line that is parallel to segment BC and let the intersection of this line and our circle be X'. |  |
| Let X' be the point at the intersection of the circle with center X and the line parallel to BC passing through X. |  |
| Now we have that XX' is also a radius of our circle, so AX=XY=XX'. |  |
| Next, construct a line that is parallel to XY and that passes through X'. |  |
| Construct a circle with center at X' and radius of XX'. Let Y' be the intersection of this parallel line and circle. |  |
| Since XX' and X'Y' are both radii of the circle centered at X' and XX'=XY=AX, then XX'=XY=AX=X'Y'. |  |
| Now construct a circle with center at Y' and radius of X'Y'. |  |
| 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'. |  |
Construct one more circle with Y' as its center and its radius being X'Y'. |
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Now, we can see that AX = XY = YB if Y=C. We must move our X until Y=C. |
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Once X is moved until Y=C, then we have finished our construction and have AX = XY = YB.
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