Let BC be chord of contact of tangents from point A to circle x^2 + y^2=1. P is any point on arc BC.

Let Px, Py ,Pz be lengths of perpendicular from P to AB,BC,AC then Px,Py Pz are in

1. A.P.
2. G.P.
3. H.P.
4. NOT

To tackle the problem involving the circle defined by the equation \(x^2 + y^2 = 1\) and the tangents from a point \(A\) to the circle, we need to delve into some properties of tangents and the relationships between points and lines in geometry. The key here is to understand the configuration of points and how the perpendicular distances from point \(P\) to the sides of triangle \(ABC\) relate to each other.

Understanding the Geometry

First, let's clarify what we have. The circle \(x^2 + y^2 = 1\) is centered at the origin with a radius of 1. The chord \(BC\) is the line segment connecting the points where the tangents from point \(A\) touch the circle. The point \(P\) lies on the arc \(BC\) of the circle, and we are interested in the perpendicular distances from \(P\) to the lines \(AB\), \(BC\), and \(AC\).

Perpendicular Distances and Their Relationships

Let’s denote the lengths of the perpendiculars from point \(P\) to the sides of triangle \(ABC\) as follows:

• \(Px\) - the distance from \(P\) to line \(AB\)
• \(Py\) - the distance from \(P\) to line \(BC\)
• \(Pz\) - the distance from \(P\) to line \(AC\)

We want to determine the relationship between these distances. The problem states that \(Px\), \(Py\), and \(Pz\) are in arithmetic progression (A.P.), geometric progression (G.P.), or harmonic progression (H.P.). To analyze this, we can use the properties of triangles and the concept of the area.

Area and Progressions

The area of triangle \(ABC\) can be expressed in terms of the perpendicular distances from point \(P\) to the sides of the triangle. The area \(A\) can be calculated using the formula:

\(A = \frac{1}{2} \times \text{base} \times \text{height}\)

For each side of the triangle, we can express the area as:

• \(A = \frac{1}{2} \times AB \times Px\)
• \(A = \frac{1}{2} \times BC \times Py\)
• \(A = \frac{1}{2} \times AC \times Pz\)

From these equations, we can see that the ratios of the areas corresponding to each side are directly related to the lengths of the perpendiculars. If we denote the lengths of the sides \(AB\), \(BC\), and \(AC\) as \(c\), \(a\), and \(b\) respectively, we can set up the following relationships:

\(c \cdot Px = a \cdot Py = b \cdot Pz\)

Analyzing the Progressions

Now, if we assume \(Px\), \(Py\), and \(Pz\) are in A.P., we would have:

\(2Py = Px + Pz\)

For G.P., the relationship would be:

\(Py^2 = Px \cdot Pz\)

For H.P., the relationship is given by:

\(\frac{1}{Px} + \frac{1}{Pz} = \frac{2}{Py}\)

However, due to the nature of the triangle and the circle, the distances \(Px\), \(Py\), and \(Pz\) do not satisfy these conditions simultaneously. The perpendicular distances from a point on the arc of a circle to the sides of a triangle formed by the tangents from an external point do not maintain a consistent ratio that would allow them to be in A.P., G.P., or H.P. This is primarily because the angles and the lengths of the sides of triangle \(ABC\) change as point \(P\) moves along the arc, disrupting any potential progression.

Conclusion

In summary, the lengths \(Px\), \(Py\), and \(Pz\) do not form an arithmetic, geometric, or harmonic progression due to the dynamic nature of the triangle and the circle's geometry. Each distance is influenced by the specific configuration of the triangle and the position of point \(P\) on the arc \(BC\). Thus, the answer to your question is that \(Px\), \(Py\), and \(Pz\) are not in A.P., G.P., or H.P.