Question

A straight line through the vertex \(P\) of a triangle \(P Q R\) intersects the side \(Q R\) at the point \(S\) and the circumcircle of the triangle \(P Q R\) at the point \(T\). If \(S\) is not the centre of the circumcircle, then (A) \(\frac{1}{P S}+\frac{1}{S T}<\frac{2}{\sqrt{Q S \times S R}}\) (B) \(\frac{1}{P S}+\frac{1}{S T}>\frac{2}{\sqrt{Q S \times S R}}\) (C) \(\frac{1}{P S}+\frac{1}{S T}<\frac{4}{Q R}\) (D) \(\frac{1}{P S}+\frac{1}{S T}>\frac{4}{Q R}\)

Short answer

(B) \(\frac{1}{P S}+\frac{1}{S T}>\frac{2}{\sqrt{Q S \times S R}}\)

Step-by-step solution

Understand the Problem

We are given a triangle with a circumcircle and a line intersecting the triangle and the circumcircle at specific points. We need to find the correct inequality involving the segments of the triangle and the circumcircle.

Relate Segments of Triangle to Circle Properties

By the Power of a Point Theorem, the product of the lengths of the two segments from S to P and S to T (the endpoints of the line through the circle) is equal to the product of the lengths from S to Q and S to R. Hence, we have the equality: SP * ST = SQ * SR.

Use Inequality Between Harmonic Mean and Geometric Mean

Since S is not the centre of the circumcircle, PS and ST are not equal, so we can apply the inequality of the harmonic mean (HM) and geometric mean (GM) for two positive unequal numbers a and b: (2ab)/(a + b) < sqrt(ab).

Relate HM and GM Inequality to Given Lengths

Using the Power of a Point Theorem result, let PS = a and ST = b, we have (2ab)/(a + b) < sqrt(ab) which translates to (2 * PS * ST) / (PS + ST) < sqrt(PS * ST). Replacing PS * ST with SQ * SR from the Power of a Point Theorem, we get (2 * SQ * SR) / (PS + ST) < sqrt(SQ * SR).

Rationalize the Inequality

Divide both sides by (SQ * SR) and multiply by 2, we get 2/(PS + ST) < 2/sqrt(SQ * SR).

Use the Reciprocal Property

If 2/(PS + ST) < 2/sqrt(SQ * SR), then the reciprocal inequality must hold true, i.e., (1/PS) + (1/ST) > 2/sqrt(SQ * SR).

Conclusion

Since we have found that (1/PS) + (1/ST) > 2/sqrt(SQ * SR), option (B) is the correct answer.

Key concepts

Circumcircle of a Triangle

The concept of a circumcircle of a triangle is vital to understanding many properties in geometry. In essence, the circumcircle is the unique circle that passes through all three vertices of a given triangle. Every triangle has a circumcircle, and the center of this circle is called the circumcenter, which is equidistant from all the vertices of the triangle.

One interesting property of the circumcircle is that it can be constructed by perpendicularly bisecting each side of the triangle and then finding the point where these bisectors intersect. This center is not necessarily within the triangle, especially in the case of obtuse triangles, where it lies outside.

In the context of the Power of a Point Theorem, a line drawn from any vertex through the circumcircle creates segments that hold special relationships. The Theorem helps us relate the lengths of these segments through a simple yet powerful equality that becomes a cornerstone in solving many geometric problems.

Harmonic Mean

Understanding the harmonic mean (HM) is essential when dealing with averages that involve rates or ratios. HM is defined as the reciprocal of the arithmetic mean of the reciprocals of a set of numbers. More simply, for two numbers, a and b, the harmonic mean is \(2ab/(a+b)\).

This concept becomes particularly useful when the relationship between two segments within a geometric figure is inverse, as is often encountered in geometric proofs and problem-solving. In our exercise, the harmonic mean provides a critical step in establishing the relationship between the segments formed by a line intersecting the triangle and its circumcircle.

Moreover, the harmonic mean is always the smallest compared to geometric and arithmetic means, except when all the numbers are equal. This understanding leads to a comparison between harmonic and geometric means that helps identify the correct inequality in geometric scenarios.

Geometric Mean Inequality

The geometric mean inequality is an insightful concept that speaks to the relationship between the product and the arithmetic mean of numbers. Formally, it states that for non-negative numbers, the geometric mean will always be less than or equal to the arithmetic mean. For two positive numbers, a and b, the inequality is expressed as \(\sqrt{ab} \leq (a+b)/2\), with equality holding only if \(a = b\).

When applied to geometry, and specifically in our exercise, this inequality becomes a tool to prove relationships between line segment lengths. The geometric mean inequality is applied to segments created by intersecting lines and circles to derive meaningful comparisons. By using inequalities like these, complex geometrical properties can be understood and resolved in a more cohesive and comprehensible manner.

It is important to note when considering the geometric mean inequality in geometric proofs, one must ensure that the segments or terms being compared adhere to the conditions of the inequality: non-negative and real.