Question

In a triangle PQR, two points S and T are on the sides PQ and PR respectively such that \(\frac{PS}{SQ} = \frac{PT}{TR}\) and \(\angle PST = \angle PRQ\), then prove that \(\triangle PQR\) is an isosceles triangle.

Hint:

The logical flow is crucial in such proofs. The ratio condition leads to parallel lines. Parallel lines lead to equal corresponding angles. Combining this with the given angle condition proves two angles of the large triangle are equal, which in turn proves it is isosceles.

Answer 1

Step 1: Understanding the Concept:
This proof involves two key theorems from triangle geometry. First, the Converse of the Basic Proportionality Theorem (also known as Thales's Theorem or BPT), which relates side ratios to parallel lines. Second, the property of isosceles triangles, which relates equal angles to equal opposite sides.

Step 2: Detailed Explanation:
We are given a \(\triangle PQR\) with points S on PQ and T on PR.
Given Condition 1: The sides are divided in the same ratio. \[ \frac{PS}{SQ} = \frac{PT}{TR} \] By the Converse of the Basic Proportionality Theorem, if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.
Therefore, we can conclude that \(ST \parallel QR\).
Now, since \(ST \parallel QR\) and PQ is a transversal line, the corresponding angles are equal. \[ \angle PST = \angle PQR \text{--- (i)} \] Given Condition 2: We are also given an equality of angles. \[ \angle PST = \angle PRQ \text{--- (ii)} \] From equations (i) and (ii), we can see that both \(\angle PQR\) and \(\angle PRQ\) are equal to \(\angle PST\). Therefore, \[ \angle PQR = \angle PRQ \] In \(\triangle PQR\), we have now shown that two angles are equal. In any triangle, the sides opposite to equal angles are equal in length.
The side opposite to \(\angle PQR\) is PR.
The side opposite to \(\angle PRQ\) is PQ.
Therefore, \(PQ = PR\).
A triangle with two equal sides is an isosceles triangle.

Step 3: Final Answer:
Thus, \(\triangle PQR\) is an isosceles triangle. Hence Proved.