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Question
ΔPBC, ΔQBC and ΔRBC are three isosceles triangles on the same base BC. Show that P, Q and R are collinear.
Solution
Given: Three isosceles triangles PBC, QBC and RBC on the same base BC such that PB = PC, QB = QC and RB = RC.
To prove: P, Q, R are collinear.
Proof: Let l be the perpendicular bisector of BC. Since, the locus of points equidistant from B and C is the perpendicular of the segment joining them. Therefore,
ΔPBC is an isosceles
⇒ PB = PC
⇒ P lies on l ...(i)
ΔQBC is isosceles
⇒ QB = QC
⇒ Q lies on l ...(ii)
ΔRBC is an isosceles
⇒ RB = RC
⇒ R lies on l ...(iii)
From (i), (ii) and (iii), it follows that P, Q and R lie on L.
Hence, P, Q and R are collinear.
Hence proved.
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