Metric geometry : Problema fundamental de tangencias : PPr

Classically tangencies problems have been studied looking geometric constructions of each case study.

Concepts power of a point on a circle can address the problems with a unifying approach, so that any tangency or incidences statement generally be reduced to a more generic fundamental problem tangents denominate (PFT).

The PFT can be stated as the problem of determining a circle passing through two points and is tangent to a line or to another circle.

A higher level of abstraction would replace items passing through a condition of belonging to a beam, although this approach will leave it pending for later.

We will solve the first case study stating the problem as:

Determine the circles through the points A and B and the line tangent to r

Data to define the fundamental problem of tangents

Analysis of the fundamental problem of tangents

In figure analysis shows that the circumference c center C can be one solution to the problem as it passes through points A and B and is tangent to the line r. In this figure ,in which the circumference represent solution we are looking for, we can determine properties that serve to derive a construction that allows us to determine it.

Basics fundamental problem of tangents PFT

The line through the points A and B the short straight r at a point P. The potency of this point on the circumference is:

Power point

From the above expression we deduce that if we get the segment value PT (Power root) we get the point T tangent and the problem reduces to determining the circle through three points: A, B and T (its center will be at the intersection of two bisectors).

Resolving the problem.

Determine the value of the power by one of the constructions used to solve proportional means:

• Height theorem
• Theorem catheter
• Power

As the power point P with respect to any circle through the points A and B is the same, We can use an auxiliary circle of any radius passing through these points, as shown in the center figure O1, located on the bisector of A and B.

The power value obtaining determine the tangent segment from P this auxiliary circle; for this, build a arc able 90 grados the segment PO1

Resolution fundamental problem of tangents

The segment value tangency ( P-T1) take it on the line r to determine the point T of tangency with a simple twist of center P.

PFT Solution

Solution Number

Depending on the direction in which we take the segment PT get one or other of the two possible solutions.

Two solutions

Metric geometry