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Question Number 218410 by Spillover last updated on 09/Apr/25

	Answered by Nicholas666 last updated on 09/Apr/25

	$g i v e n; A B, C D t a n g e n t s t o c i r c l e O :$ $M m i d p o i n t o f B C; A D ∣∣ B C; B D = x, A B = y, C D = z$ $p r o v e x^{2} = y . z$ $s o l u t i o n$ $* A D ∣∣ B C ∠ A B D = ∠ D B C (a l t e r n a t e i n t e r i o r a n g l e s)$ $* ∠ A B D = ∠ B D C (a n g l e b e t w e e n t a n g e n t a n d c h o r d)$ $* A B D \sim △ B D C (A A : ∠ A B D = ∠ D B C, ∠ A B D = ∠ B D C)$ $* r a t i o o f c o r r e s p o n d i n g s i d e s; A B / B D = B D / C D$ $* y / x = x / z = y . z = x . x = y . z = x^{2}$ $* c r o s s - m u l t i p l y : x^{2} = y . z$ $q . e . d$
	Answered by Spillover last updated on 10/Apr/25

	Answered by Spillover last updated on 10/Apr/25

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