Unizor - Geometry2D - Similarity - Construction 3
Channel: Zor Shekhtman on UNIZOR Education
Date: 14/08/2012

1. Construct segments expressed by the following formulas:
(a, b, c and d are given segments;
x is a segment we want to construct;
m, n are natural numbers;
unit segment of length 1 is always considered as given)
x = a·n
x = a/n
x = a·(m/n)
x = a·b/c
x = a/b [means x = a·1/b]
x = a2/b
x = a·(m2)
x = √a·b+c·d 
x2 = a·b
x = √m  [means x2 = m·1]

2. Calculate the length of a secant to a given circle from a given point if a tangent from this point has a length a and a ratio between internal and external parts of a secant is m:n.

3. Construct a segment x with a length having a ratio to a length of a given segment s equal to a ratio of squares of lengths of given segments p and q. 
In short, x : s = p^2 : q^2.

4. Find a point outside of a given circle such that a tangent from it to this circle is measured as half a secant from it through a center of a circle to an opposite endpoint of a diameter.

5. Construct a secant from a given point to a given circle that is divided by this circle in a given ratio m:n.

6. Given two circles of different radii a and b and a distance between their centers d. Express as an algebraic formula the length of a segment from a point of intersection between a center line and common exterior tangent (the one with both circles positioned on one side of it) to a nearest circle. Construct a segment according to this formula.