Let AB be diameter of circle \\omega and let C be point on \\omega, different from A and B. The perpendicular from C intersects AB at D and \\omega at E( C). The circle with centre at C and radius CD intersects \\omega at P and Q. If the perimeter of the triangle PEQ is 24, find the length of the side PQ.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. CPEQ is cyclic quadrilateral.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. CP × EQ + CQ × PE = CE × PQ.
Hint 3: Proceed with the final algebraic steps to solve the system. r(EQ + PE) solve for the final valuer PQ.
Step 1: CPEQ is cyclic quadrilateral
Step 2: CP × EQ + CQ × PE = CE × PQ
Step 3: r(EQ + PE) = 2r PQ
Step 4: 2PQ = EQ + PE
Step 5: PQ + EQ + PE = 24
Step 6: PQ + 2PQ = 24
Step 7: => PQ = 8
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