Twin triangles!

Geometry Level 3

If in Δ P Q R \Delta PQR , L L and M M are points on Q R QR such that ∠ L P Q = ∠ Q R P \angle LPQ = \angle QRP and ∠ R P M = ∠ R Q P \angle RPM = \angle RQP , then which of the following is P Q 2 PQ^2 equal to?

Q R × M R QR \times MR Q R × L R QR \times LR Q R × Q L QR \times QL Q R × Q M QR \times QM

Ashrit Ramadurgam by Ashrit Ramadurgam , 20, India

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1 solution

Ashrit Ramadurgam
Mar 20, 2016

∠ L P Q = ∠ Q R P ⋯ A \angle LPQ= \angle QRP \cdots A ∠ R P M = ∠ R Q P ⋯ A \angle RPM= \angle RQP \cdots A By Angle-Angle similarity criterion, Δ P Q L ∼ Δ P M R \Delta PQL \sim \Delta PMR ⇒ P Q Q R = Q L P Q \Rightarrow \frac{PQ}{QR} = \frac{QL}{PQ} P Q 2 = Q R × Q L PQ^2 = QR \times QL

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