Proof:

Given: A, B, and C are three points on a line, and B lies between A and C.

To prove: AB * BC = AC

Proof by Euclid's Elements:

Statement 1: Consider a line segment AB.

Reason: By definition, AB is a line segment.

Statement 2: Consider a line segment BC.

Reason: By definition, BC is a line segment.

Statement 3: Since B lies between A and C, A, B, and C are collinear.

Reason: By definition, collinear points lie on the same line.

Statement 4: By the Segment Addition Postulate, AB + BC = AC.

Reason: The Segment Addition Postulate states that if three points A, B, and C are collinear and B is between A and C, then AB + BC = AC.

Statement 5: Multiplying both sides of the equation AB + BC = AC by BC, we get AB * BC + BC * BC = AC * BC.

Reason: Multiplying both sides of an equation by the same value maintains equality.

Statement 6: BC * BC can be simplified to BC^2.

Reason: Multiplying a number by itself is equivalent to squaring it.

Statement 7: Simplifying the equation AB * BC + BC^2 = AC * BC, we get AB * BC = AC * BC - BC^2.

Reason: Subtracting BC^2 from both sides of the equation.

Statement 8: Simplifying further, AB * BC = BC * (AC - BC).

Reason: Factoring out BC from the right side of the equation.

Statement 9: Since B lies between A and C, AC - BC = AB.

Reason: By the Segment Subtraction Postulate, if B is between A and C, then AC - BC = AB.

Statement 10: Substituting AB for AC - BC in the equation AB * BC = BC * (AC - BC), we get AB * BC = BC * AB.

Reason: Substituting the value of AC - BC with AB.

Statement 11: AB * BC = BC * AB can be simplified to AB * BC = AB * BC.

Reason: Since BC and AB are equal, multiplying AB by BC or BC by AB will yield the same value.

Statement 12: AB * BC = AB * BC is a true statement.

Reason: The equation is symmetric, and both sides are equal.

Therefore, it is proved that AB * BC = AC.