history of mathematics - National STEM Centre

76
Descartes
Figure 6.5
Figure 6.5 shows that there are other points, not on AB, that lie on a curve that looks
as though it may be a circle. After drawing a few of these points you may be ready
to make a hypothesis.
Hypothesis
Figure 6.6
The set of points X that satisfy the condition that AX = 2BX is a circle, as shown in
Figure 6.6. The centre M of the circle is on AB produced, such that MB = ^ AB, and
the radius is -| AB.
Now that you have investigated the problem and made a hypothesis, try to prove
that your hypothesis is correct.
Proof
Figure 6.7
For each point X on the circle shown in Figure 6.7
• ZAMX=ZXMB
• AM : XM = 2 :1 (since AM = AB + BM = JAB and XM = radius = f AB)
• also MX : MB = 2 :1 (why?)
So the triangles AMX and XMB are similar. Therefore AX : XB = 2 :1.