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

AandBand the line tangent tor

## 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.

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:

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:

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**

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**.

## Solution Number

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

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