Question

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

Hint:

Reciprocal Roots Condition: If one root is reciprocal of the other, use \( \alpha \beta = 1 \).

Answer 1

Since \( \frac{PS}{SQ} = \frac{PT}{TR} \), by the Basic Proportionality Theorem,
\[ ST \parallel QR \] Since \( \angle PST = \angle PRQ \), corresponding angles are equal, proving \( \triangle PST \sim \triangle PRQ \).
From similarity:
\[ \frac{PS}{PQ} = \frac{PT}{PR} \] Since proportions hold, \( PQ = PR \), proving \( \triangle PQR \) is isosceles.
Correct Answer: \( \triangle PQR \) is an isosceles triangle.