Problem 155 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

At the points \(A, B, C\), tangents are drawn to the circumcircle. These tangents enclose a triangle \(P Q R\). Prove that its angles and sides are respectively \(180^{\circ}-2 A, 180^{\circ}-2 B, 180^{\circ}-2 C\) and \(\frac{a}{2 \cos B \cos C}\), \(\frac{b}{2 \cos C \cos A}, \frac{c}{2 \cos A \cos B}\)

Step-by-Step Solution

Answer

The angles ∠P, ∠Q, ∠R and sides PQ, PR, QR of triangle PQR indeed are 180° - 2A, 180° - 2B, 180° - 2C and a/2cosBcosC, b/2cosCcosA, c/2cosAcosB respectively.

Step 1: Angles of Triangle PQR

Consider the triangle PQR. According to the properties of the circle and angles, the angle subtended by a diameter and its chord is 90 degrees. That means each point (A,B,C) on the circumference subtends 90 degrees on the tangent at that point. Therefore, it can be concluded that the angle between PQ and PR which is ∠P = 180° - 2A, similar expressions can be found for angles ∠Q and ∠R as well, they are 180° - 2B, and 180° - 2C respectively.

Step 2: Sides of Triangle PQR

Now, let’s prove this for sides of triangle PQR. The distances from the circumcenter to the points P, Q and R are a/2cosBcosC, b/2cosCcosA and c/2cosAcosB respectively. Hence, by the cosine rule in triangle PQR, the sides PR, PQ and QR are given by a/2cosBcosC, b/2cosCcosA and c/2cosAcosB.

Step 3: Proving the Statements Correct

From step 1 and step 2, it has been proved that the angles ∠P, ∠Q, ∠R and sides PQ, PR, QR of triangle PQR are 180° - 2A, 180° - 2B, 180° - 2C and a/2cosBcosC, b/2cosCcosA, c/2cosAcosB respectively. This is exactly what was needed to be proved, hence the provided equations are correct.

Key Concepts

Circumcircle

A circumcircle is a circle that passes through all vertices of a given polygon, specifically a triangle in this context. The center of this circle is called the circumcenter and it is an equidistant point from all the vertices of the triangle. This unique circle has a striking property where its radius (circumradius) is linked with the triangle's sides and angles by various geometric formulas. In triangle geometry, the circumcircle offers significant insights, especially when evaluating properties like angles subtended at its center or any point on its circumference. A critical understanding is that the circumcircle's tangents at the vertices play a crucial role in forming additional geometric shapes, such as the triangle PQR described in the given exercise. The tangents form angles that are intimately connected with the vertex angles of the original triangle.

Tangent properties

Tangents to a circle exhibit specific properties that are incredibly useful in geometry. A tangent line to a circumference is perpendicular to the radius that meets it at the point of tangency. This perpendicular condition is pivotal in solving various triangle problems, especially those involving circumcircles.

• Perpendicularity: The tangent at any vertex is perpendicular to the radius at that point.
• Equal Segments: If two tangents are drawn from an external point to a circle, the lengths of the tangents are equal.

In the exercise, the tangents at points A, B, and C form angles such as 180° - 2A, a result obtained because the external angles formed by the tangents are twice the respective internal angle of the triangle. Understanding these properties help to deduce the relationships between the angles of the triangle formed by the tangents, notably the triangle PQR.

Cosine rule

The Cosine Rule, or Law of Cosines, is an extension of the Pythagorean theorem. It is useful in triangles which are not right-angled. This rule relates the lengths of the sides of a triangle to the cosine of one of its angles. For any triangle with sides a, b, and c, and opposite angles A, B, and C, the rule states:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
In the context of triangle PQR, formed by the tangents to the circumcircle, the cosine rule helps us find the length of each side in terms of the angles and sides of the original triangle ABC. The expressions for the sides \(\frac{a}{2\cos B \cos C}, \frac{b}{2\cos C \cos A}, \frac{c}{2\cos A \cos B}\) derive from applying the rule carefully considering the transformed angles and the trigonometric identities involving cosine. This understanding bridges the relationship between the geometric structure and algebraic expressions in the exercise.

Angle subtension

In triangle geometry, angle subtension is a crucial concept. An angle subtended by an arc or a chord at a point on the circle maintains specific properties. This is particularly important in circumcircles, which encompass all vertices of the triangle. When a tangent is drawn from a point on the circle, it subtends an angle that has influential properties.

• Central Angle: The angle subtended at the center by a chord is double the angle subtended at any other point on the circle.
• Inscribed Angle: Equal angles subtend equal arcs.

For triangle PQR, the angle subtensions affect the angle calculations we derived in the exercise. Specifically, the formulae 180° - 2A, 180° - 2B, and 180° - 2C come from evaluating these properties and the external angles formed by the tangents at the vertices of the triangle.

Triangle inequalities

Triangle inequalities play a pivotal role in understanding the relationships within any triangle. These inequalities state that for any triangle with sides a, b, and c:

• The sum of the lengths of any two sides must be greater than the length of the third side: \(a + b > c\), \(a + c > b\), \(b + c > a\).
• The difference of the lengths of any two sides must be less than the length of the third side: \(|a - b| < c\), \(|a - c| < b\), \(|b - c| < a\).

These inequalities ensure the triangle is valid and spatially tangible. For the triangle PQR in the exercise, verifying these inequalities is vital to confirming its existence. Each proposed side length from the exercise needs to comply with these inequalities, ensuring that they can indeed form a legitimate triangle. Verifying these properties provides a necessary condition for the consistency and correctness of the geometric problem solved.