P is the centre of the circle. Prove that $∠ X P Z = 2 (∠ X Z Y + ∠ Y X Z)$ .

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Solution

Given : A circle with centre P, XY and YZ are two chords.
To prove : $∠ X P Z = 2 (∠ X Z Y + ∠ Y X Z)$
Proof : In a circle with centre P, arc XY subtends $∠ X P Y$ at the centre and $∠ X Z Y$ at remaining part of the circle.
$\Rightarrow ∠ X P Y = 2 ∠ X Z Y$ ....(1)
Similarly, arc YZ subtends $∠ Y P Z$ at the centre and $∠ Y X Z$ at remaining part
$∴ ∠ Y P Z = 2 ∠ Y X Z$ ....(2)
Adding (1), and (2), we get
$∠ X P Y + ∠ Y P Z = 2 ∠ X Z Y + 2 ∠ Y X Z$
$\Rightarrow ∠ X P Z = 2 (∠ X Z Y + ∠ Y X Z)$ .

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