Question

The points A(x1, y1) and B(x2,y2) are two points on the graph of y = log(x). Through the midpoint of the line segment AB, a horizontal line is drawn to cut the curve at C(x3,y3).
Prove that (x3)^2 = x1x2

Answer 1

\[
y_3= \frac {y1+y2}2
\]

Answer 2

\[
x3= e^{y3}= e^{\frac{ y1+y2}2}
\]

Answer 3

\[x3^2 = e^{y1+y2}= e^{\ln x1 + \ln x2}= e^{\ln (x1 x2}= x1 x2
\]